Casting a recursive enough Thoth's Gambit can go forever; for trivial internal Patterns, I've got ~42k iterations of the loop on my local machine before a timeout occurred and booted me or crashed the server. There's a variety of ways to achieve loops that would go infinitely were those timeouts not to happen first (eg, a focus with a pattern including its a repeating Scribe's Reflection being fed into a Thoth).

Looking at the code, OpEval (Hermes?) seems to be the only place where depth for a frame is increased. I'm not sure what the intended limits for Thoth's Gambit are supposed to be, though; limiting its max loop size to 64 (minus any other Hermes or Thoth occurring in the same spell) seems to make it very weak, but it's pretty trivial to come up stack exploitation for Hermes that could hit 50k+ actions without requiring more than two or three levels of actual call depth.