Take a random walk in the plane from the origin, each step of unit length in a uniformly random direction.
Q. How many steps on average until the path self-intersects?
My simulations suggest ~$8.95$ steps.
Red: origin. Top: the $8$-th step self-intersects. Bottom: the $11$-th step self-intersects. (Not to same scale.)
I suspect this is known in the SAW literature (SAW=Self-Avoiding Walk), but I am not finding this explicit number.
Related: self-avoidance time of random walk.
Added. Here is a histogram of the number of steps to self-intersection.
$10000$ random trials.