Does the code terminate?
Jelly, 7 bytes

!‘Æ²µ4#


Try it online!

Background

This will terminate once it finds a fourth solution to Brocard's problem, i.e. a solution n! + 1 = m² with (n,m) ≠ (4, 5), (5, 11), (7, 71) over the positive integers. The implementation doesn't use floating point arithmetic, so it will only terminate if it does find a fourth solution or if n! can no longer be represented in memory.

Brocard's problem was first used in this answer by @xnor.

How it works

!‘Æ²µ4#  Main link. No arguments. Implicit argument: 0

    µ4#  Convert the links to the left into a monadic chain and call it with
         arguments k = 0, 1, 2, ... until 4 of them return 1.
!        Factorial; yield k!.
 ‘       Increment; yield k! + 1.
  Æ²     Squareness; return 1 if k! + 1 is a perfect square, 0 if not.