Can a combinatorial hardware circuit with no inputs solve a crypto problem?
The general field of digital logic without a clock is asynchronous logic. Research has been done in this area but it hasn't really been commercialised.

Your cryptographic example might fail for two reasons:


it may find a cycle, possibly a long one, that produces a sequence of values but does not converge on the desired solution. This is a mathematical aspect of the problem rather than an implementation detail.
If you use regular logic and don't have some system for keeping signal propagation in sync, you won't compute the right thing at all, as some parts of the answer will be ready before others, which in turn will perturb the input. See the "Muller C-element" for a solution to this.


Your engineer should have told you that you were reinventing the "ring oscillator". These are often used in conjunction with phase-locked-loops to produce high frequency digital clocks that are some multiple of an input clock without using a high-frequency crystal.