I'm trying to come up with an example of a curve that has constant curvature and torsion, both exactly 1.
Let $\vec{r(t)}$ be a parametrization for the curve $\gamma$. By calculating with formulas for curvature $\kappa$ and torsion $\tau$ I got that the length of the second derivative $\dot{\vec{r}} $ must be the square of the length of the first derivative. Similarly, I got that the length of the third derivative must be a cube of the first derivative. But I couldn't get this any further.
I'm guessing it could be something like a spiral... that would give us a constant curvature. And this spiral should bend...
What would be such an example?
(For definition of the curvature and torsion: https://en.m.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas)
Your intuition is correct. Consider the unit-speed helix $$\beta(s)=\left(a\cos\frac{s}{c}, a\sin\frac{s}{c},\frac{bs}{c}\right),$$ where $c=\sqrt{a^2+b^2}$ and $a>0.$ You can check that $$\kappa(s)=\frac{a}{a^2+b^2},$$ and $$\tau(s)=\frac{b}{a^2+b^2}.$$ In particular, if $a=b=1/2,$ then both $\kappa$ and $\tau$ are equal to $1$.