Are there any NP-complete problems where the fastest known algorithm solves the problem in superexponential time (like $O(n!)$ time)? Every NP-complete problem that I am aware of has fastest known algorithms that are all exponential time, but it is not obvious to me whether there is anything preventing such problems from having best-case algorithms that run in superexponential time.
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I’m not sure how you define “superexponential”, could you make it precise for me?
If you define "super-exponential" as something described in the previous link like $2^{n^c}$: In this context, $O(n!)$ (as you described) is something bounded by $O(n^n) = O(2^{n\log n}) \leq O(2^{n^2})$, not super-exponential; and since $\sf NP \subseteq EXP$, all problems in $\sf NP$ should be bounded by this "exponential time".
If you define "super-exponential" traditionally, i.e. $2^{O(n)}$: In this case, if you can find a $\sf NP$-complete problem $A$ that is not in ${\sf TIME}(2^{O(n)})$, then it's not in ${\sf SPACE}(n)$ either, thus we can know ${\sf NP} \nsubseteq {\sf SPACE}(n)$; but it's still unknown that if ${\sf NP} \subseteq {\sf SPACE}(n)$ now (maybe new results are published? I don't know, and I learn this from CMU 15-455 hw7 p4, spring 2017 by Prof. Ryan), so I guess proving this is hard.
(In fact this is more like a comment than an answer, but I don’t have enough reputation to do so, sorry about that)
Heda Chen
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