$$
\newcommand{\vvec}{{\bf{v}}}
\newcommand{\wvec}{{\bf{w}}}
\newcommand{\Mmat}{{\bf{M}}}
$$
You can quite easily change the metric on the unit interval. The metric at a point $t$ of the manifold in this case is specified by a $1 \times 1$ symmetric positive definite matrix $\Mmat(t)$, with the inner product of two vectors $\vvec$ and $\wvec$ in the tangent space at $t$ being given by $\vvec^t \Mmat(t) \wvec$. Since $\vvec$ is just a number, and so is $\wvec$, this turns into a product of three numbers $vmw$, where $m$ is a positive number.
As an example, we can try something like $\Mmat(t) = 1/{t^2}$. Now the vector $[1]$ at $t = \frac{1}{4}$ has squared length $4$, but at $t = 1$, it has squared length $[1][1][1] = 1$, hence length $1$ as well. (For this to make sense, I'd need to extend your interval to, say, $(0, 3/2)$, so that $1$ is in the manifold...sorry about that!)
What's a geodesic look like here? It'll still be a curve traversing the unit interval, but in the usual coordinates on $\mathbb R$, it'll have to speed up as it goes from $0$ to $1$. In fact, $\gamma(t) = e^t$ is, I believe, a geodesic in this metric. Let's check that.
The speed of $\gamma$ is the length of $\dot{\gamma}$. Let's look at a particular $t$, say $t_0 = 1/4$. There, we have the following data:
\begin{align}
\gamma(t_0) &= e^{0.25}\\
\dot{\gamma}(t_0) &= [e^{0.25}]\\
\Mmat(\gamma(t_0)) &= [\frac{1}{\gamma(t_0)^2}] = [\frac{1}{e^{0.5}}]\\
\|\dot{\gamma}(t_0) \|^2 &= \dot{\gamma}(t_0)^t \Mmat(\gamma(t_0)) \dot{\gamma}(t_0) \\
&= [e^{0.25}]^t [\frac{1}{e^{0.5}}] [e^{0.25}]^t \\
&= [1]
\end{align}
Sure enough, it's a unit-speed curve at that point (and at other points, as I'm sure you can verify).
Great question!
To continue, answering a followup from the comment, we can work through the computation of the Christoffel symbols (there's only 1) at the point $t$:
\begin{align}
\Gamma_{11}^1 (t) &= \frac{1}{2} g^{11} (\frac{\partial g_{11}}{\partial t} + \frac{\partial g_{11}}{\partial t} - \frac{\partial g_{11}}{\partial t}) \\
&= \frac{1}{2} g^{11} (\frac{\partial g_{11}}{\partial t}) \\
&= \frac{1}{2} t^2 (\frac{-2}{t^3}) \\
&= \frac{1}{t}.
\end{align}
Note that I've written $\Gamma_{11}^1 (t)$ to indicate that this is a function that varies with the coordinate of the point you're considering. (I've also computed $g^{11} = t^2$, the sole entry of the inverse of the matrix $g_{11} = \frac{1}{t^2}$.)
The equation for geodesics is
$$
\frac{d^2 t}{d s^2}(s) + \Gamma_{11}^1 (t(s)) \frac{dt}{ds}(s)\frac{dt}{ds}(s) = 0
$$
where the curve is given by
$$
s \mapsto t(s).
$$
In our case, the curve is $s \mapsto e^s$. So the equation becomes
$$
e^s + \frac{1}{e^s} \left( e^s e^s\right) = 0
$$
which turns out to be true for all $s$. Hence our curve is a geodesic.
There are many alternative characterizations of geodesics, like "locally shortest paths that are unit speed," which is how I knew we were done when I had
"unit speed", since in a 1-manifold there's only one direction to go (up to sign). But actually checking the geodesics equation is probably a Good Thing to do as well, just for practice. (I confess, I think I got it wrong 3 times before writing this down.)
