Puffy Jupiters can be huge but the Kepler data suggests that traditional super-Jovian worlds nearer the goldilocks zone, or farther from their parent star, only reach about 10% larger in diameter than Jupiter to spite being 5-10 times as massive.
With that in mind what is the greatest percentage of the sky of an Earth-like habitable moon, in a stable, circular orbit, that a gas giant world 10-15% larger than Jupiter and 10 times as massive could possibly cover?
There are scientific calculations indicating that a habitable exomoon of an a giant exoplanet would have to orbit beyond what its called the "habitable edge" of that exoplanet, which has been stated to be 5 times the radius of the exoplanet.
So an almost circular orbit with 5 times the radius of the planet would have a semi-major axis that was 2.5 times the diameter of the planet. Thus that orbit would have a circumference that was about 15.70795 times the diameter of the planet, and so the planet should have an angular diameter of about 22.918 degrees.
According to this online angular diameter calculator the planet would have an angular diameter of 22.62 degrees.
https://rechneronline.de/sehwinkel/angular-diameter.php
Here is a link to the article where the term habitable edge was first used:
The abstract includes:
...We identify combinations of physical and orbital parameters for which radiative and tidal heating are strong enough to trigger a runaway greenhouse. By analogy with the circumstellar habitable zone, these constraints define a circumplanetary "habitable edge...
I note that in the case of the 2 exoplanets they cosider, they say in the abstract:
...If either planet hosted a satellite at a distance greater than 10 planetary radii, then this could indicate the presence of a habitable moon.
Of course the body of the paper is much longer than the abstract and contains much more information.
I note that if the habitable edges of those 2 specific planets are at 10 planetary radii (which would be 5 planetary diameters) the angular diameter of those planets as seen from a moon orbiting at the habitable edge would be 11.421 degrees. So is the figure of 5 plnetary radii I remember an error for 5 planetary diameters? Or does the distance of a giant planet's habitable edge vary with different factors and is 5 plentary radii thus the minimum possible habitable edge when factors allow it to be as close to the planet as possible?
Rene Heller has authoried and coauthored a number of astronomical and astrobiological papers, including over a dozen about the theoretical habitabiity of exomoons.
In "Magnetic shielding of exomoons beyond the circumplanetary habitable edge", The Astrophysical Journal, Sept. 2013, Rene Heller and Joge Zuluaga consider the case of exomoons which are larger than Mars but smaller than Earth.
Moons of that size would probably be far to small to be habitable for humans and thus habitable in a science fiction sense, but could be large enough to be habitable for some liquid water using lifeforms, and so habitable in a scientific sense.
They assume that exomoons of that size would be too small to generation strong magnetic fields, but would need to be shielded from the solar winds of their stars by magnetic fields. So they calculate the evolution of the magnetic fields of giant planets which the hypothetical exomoons might orbit, to see if the magnetic fields of the planet could extend to orbits of the moons in time to prevent them from losing all their atmospheres to the stellar winds. Could such moons be far enough fro m the planets to avoid runaway greenhouse (RG) fates while being close enough to the planets to be shielded by the planet's magnetic fields?
Their abstract says:
For modest eccentricities, we find that satellites18around Neptune-sized planets in the center of the HZ around K dwarf stars will either be in an RG state and not be habitable, or they will be in wide orbits where they will not be affected by the planetary magnetosphere. Saturn- like planets have stronger fields, and Jupiter-like planets could coat close-in habitable moons soon after formation. Moons at distances between about 5 and 20 planetary radii from a giant planet can be habitable from an illumination and tidal heating point of view, but still the planetary magnetosphere would critically influence their habitability.
And here they say that the exomoons would have to orbit between 5 and 20 planetary radii to be habitable.
I note that if you want your moon to be larger than any the solar system, but only a few times as massive as Mars, and habitable only for bacteria and single celled lifeforms, but not for humans or other large animals who need an oxygen rich atmsphere, you should have that moon orbit the planet between 5 and 20 planetary raddi, and thus the planet should have an angular diameter between 5.725 degrees and 22.62 degrees.
But if you want your habitable moon to be habitable for humans or for other lifeforms which also need an oxygen rich atmosphere, That moon can not be much less massive than Earth. And thus it will be massive enough to generate its magnetic field for shielding against stellar wind. thus such an Earth sized moon could orbit it splanet at a distance greater than 20 planetary radii and still be habitable, and thus the planet could have an angular diameter less than 5.725 degrees.
But, although it is still poorly understood how astronomical bodies generate magnetic fields, it seems to be a common opinion that worlds that spin faster are likely to have stronger magnetic fields. And your habitable moon would be tidally ocked to it's planet, and so it would a rotation period the same length as its orbital period.
The farther a tidally locked moon is from its planet, the longer its orbital period will be, and thus the longer its rotational period will, and thus probably the weaker its magnetic field will be. So you will want your human habitable moon to orbit as close to the planet as possible (but still outside the habitable edge) so its orbital period and thus rotation epriod will enable it to rotate fast enough to generate a strong magnetic field.
Anyway, the question asks about how large a giant planet could look from a habitable moon, so you will want to put your moons close to the habitable edge to make the planet have as large an angular diameter as possible. You are not interested in how far away a habitable moon could get from the planet, but how close.
I also note that in Habitable Planets for Man, 1964, Stephen H. Dole discussed the conditions necessary for a planet to be habitable for humans.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
Naturally that included a discussion of how long or short a world's day could be on pages 59-61. Dole guessed that a day 96 Earth hours of 4 Earth days long would be the longest possible day length for a habitable planet.
rom the standpoint of human Habitation, there are two limits related to rotation rates. For slow rotation rates, a limit would be reached when daytime temperatures become excessively high in the low latitudes below some critical latitude and when nighttime temperatures become excessively low polewards from this same latitude, or when the light-darkness cylcle became too slow to enable planets to live though the long hot days and long cold nights. If rotation rates were increased steadily, a limitingpoint would be reached when surface gravity at the equator fell to zero and amater was lost form the planet, or when the shape of the surface became unstable and axial symmetry was lost.
Just what extremes of rotation rate are compatable with habitability is difficult to say. These extremes, however, might be estimated at, say, 96 hours (4 Earth days) per revolution at the lower end of the scale and 2 to 3 hourss per revolution at the upper end, or at angular velocities where the shape becomes unstable due ot the high rotation rate.
My answer to the question:
Would life be possible on a planet whose poles each directly face the sun once a year?
Includes a section near the end that discusses the orbital periods and thus the rotational periods of tidally locked moons of giant planets. And it includes calculations of the orbital periods (and thus days) of moons orbiting at the habitable edges of giant planets of specified mass and diameter.
My answer to this question:
What is the largest possible appearance of a celestial body in the sky?
Discusses various posible astronomical bodies and how large they could appear from the surfaces of other astronomical bodies.
I concluded that the largest possible angular diameter of a world as seen from a habitable world would be a giant planet as seen from a habitable moon, and that its angular diameter could get up to about 22.9183 degrees.
Starfish Prime, in his answer, claims that a habitable moon could get much closer to a giant planet, and thus have a much larger angular diameter. He claims that tidal heating would only give a large moon significent heating and cuase a runaway greenhouse effect if there were other large moons in the system to give the habitable moon a more eccentric orbit.
Starfish Prime says that since the habitable moon would be denser than the giant planet it wouldn't have to worry about the Roche radius of he planet, and the two objects could orbit with just a few thousand kilometers between the upper layers of their atmospheres. He calculates a maximum possible angular diameter of 131 degrees!
The answer by L. IJspeert gives figures of 104 degrees, 65 degrees, or 19 degrees.
My answer to the question:
How would the reflection of sunlight change the day / night cycle of a binary planet system?
Involve two Earth like planets orbiting their common center of gravity just outside of their mutual Roche limit. I imagined that each would have the radius (6,371 kilometers) and diameter (12,742 kilometers) of Earth, and that there would be 2,000 kilometers between their closest points. In that case each planet would have an angular diameter of 74.584 degrees as seen from the nearest point of the other planet, and 46.745 degrees as seen from places on the other planet where it would be on the horizon.
Of course that ignores the fact that the tidal interactions between the two panets would causes their orbits to slowly change over the billions of years it would take them to become habitable for humans, moving farther apart (making them look smaller, or closer together, which could eventually destroy them.
The realistic "safe" distance of a moon from its planet can be deduced from the Roche limit !
Thanks to Wikipedia, the Roche limit is the minimum distance a large object (larger than a few tens of kilometers) can approach another object without being torn apart by the object's gravity.
Please note that the Roche limit is the minimum safe distance for a large body near a larger body in which one piece can remain, it does not express the most likely orbit of a moon around its planet, in fact, it is very rare for large moons to be in any way close to Roche limit of it's planet, moons like stable and safe orbits.
In very simple considerations that do not take into account the distribution of mass in the bodies, nor the fact that the bodies are moving, this is a good equation for the Roche limit:
$$d = R_M (2 \frac{{\rho}_M}{{\rho}_m})^{\frac{1}{3}}$$
Where d is the Roche limit radius from the center of the planet, Rm is the radius of the plane, ρm is the density of the planet, ρm is the density of the Moon, very simple isn't it?
So for a planet with 115% of the diameter of Jupiter and ten times its mass and a moon with the same size and mass as the Earth, this gives us 118000 km.
But if you ask me, I think it is better to make the value of the orbit higher than that, as the moons near the Roche limit become unstable and unlikely, especially for a moon the size of the Earth, it suffices that Io, the moon of Jupiter closest to it, has active and unstable infernal volcanoes that make it the most volcanically active place in the solar system, is all due to Jupiter's attraction and proximity to it, despite the fact that Io's orbit is seven times the Roche limit for Jupiter!!
Based on this, it can be said that a distance equal to 5 times the Roche limit of a moon makes it at the lowest distance from its planet where it does not become a ball of infernal magma, and of course this distance is improbable (but still possible) for a habitable planet the size of an Earth.
So I think this is how it looks from different orbits:
Orbit = 5 of the Roche limit = 590000 km
What a romantic and beautiful scene! Unfortunately, life on this planet has another opinion, volcanoes are terrible and bad, and the atmosphere is full of volcanic ash that blocks the sun's rays for very long periods, life is of course rare
Orbit = 7 of the Roche limit = 826000 km
What a romantic and beautiful scene, albeit a smaller one! The dangers and instability here are similar to Io, the planet has many infernal active volcanoes and earthquakes, but at least there is a good life you can live!
Orbit = 21 of the Roche limit = 2.478 million km
What a nice sight! This orbit is very stable and safe for its distance. The internal activity (volcanic and earthquakes) can be higher than the normal rate of the planets, but it is nevertheless completely habitable!
And of course we have...
Apocalypse!!
This is exactly an orbit in the limit of Roche, the planet will be destroyed and annihilated and turn into giant rings surrounding the gas giant within days!
Pictures simulation from Space Engine 0.990 !
A visual simulation of the whole universe (from the smallest asteroids to the largest galaxies!), It's one of the most difficult programs I can imagine how hard it was to make!
