I know there is a simple method that generates a sequence of closed shapes whose limit is the unit circle but the limit of the perimeter is not $2\pi$ , but in all of the cases that I know , the perimeter limit is finite.
How we can generate a sequence of shapes whose limit tends to unit circle but the limit of the perimeter is infinity?
In the annulus with inner radius $1-1/n$ and outer radius $1+1/n$ draw a curve that oscillates back and forth between the inner and outer edge often enough to have length $n$. Then the limit as $n \to \infty$ does what you ask for.
Consider the polar curve:
$$r=1+\frac1n\sin(n^2 \theta)$$
which for $n=2$ and $n=10$ looks like the following, corresponding to Ethan Bolker's suggestion.
Clearly, as $n$ increases, the curve approaches a circle of radius $1$,
though the arc length is always greater than $4n$ and so has an infinite limit.
Initially , I would have given something like what Ethan Bolker rightly gave. That Solution is already available now , so I have cooked up something even more weird.
OP is asking for the Sequence Case where the Perimeter limit is $\infty$ & Ethan Bolker has given that.
My Solution is the Sequence Case where every Element of the Sequence has Perimeter $\infty$ !
Consider the Curve $y = C x \sin (1/x-1)$ between $x = 0$ & $x = 1$.
It will look like this , with $C = 1$ :
[[ Sin Curve generated Courtesy of Wolfram Online Tool ]]
The line on the X-Axis has length $C = 1$ , while the "Sin" Curve has length $\infty$
We can "shrink" or "enlarge" the X-Axis (length) to make the length $C$ shorter or longer , while we can "shrink" or "enlarge" the Y-Axis (height) by the Same Constant $C$. The "Sin" Curve length will remain $\infty$.
We can make a Polygon ( Triangle , Square , Pentagon , Hexagon , ... n-gon , ... ) inside the Unit Circle.
Each side the Polygon can be the "Sin" Curve , rotated by the necessary angle with "shrinking" factor $C$ which is the Side length of that Polygon.
Each Side of that Polygon has length $\infty$.
Each Polygon has Perimeter $\infty$ too.
In the limit , the Side lengths of the Polygon will tend to $0$ , hence the height of the "Sin" Curves will tend to $0$ & the Polygon will tend to the Unit Circle , while the Perimeter will always remain $\infty$ for each Element of the Sequence.
Visualization : Here is the "Square" Element in that Sequence :
The Green Outline is the limiting Unit Circle , the Purple Outline is the Polygon which we are going to convert to $\infty$ length , while the Blue Curve is the Collection of "Sin" Curves which will be gradually "shrinking" to have the eventual $0$ height , while remaining with length $\infty$ . . . .
Perimeter is always $\infty$ in the Sequence here.
