As a TCS/scheduling person, this one is definitely of lesser importance than UGC, but it has been an open problem since the book of Garey and Johnson in 1979:

A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]

Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:

Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.

That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.

[1]: https://github.com/openai/math/blob/main/preprints/A-polynom...

The largest I've seen [1] is an exponent of 10^12, which I suppose still counts as polynomial time.

I'm sure all of these super small or large constants will improve over time, but it's still amusing. It is entertaining to see the exponents directly rather than have them hidden as n^c or epsilon or O(1).

[1]: https://github.com/openai/math/blob/main/preprints/Determini...