Solving an inequality $BIs there a way to solve $B<n!$ where $B$ is some very large real number (suppose for example $B=10^{17}$) without a calculator or gamma function?
At the very least, to find the nearest integer for $n$ to make the inequality true?

If you knew that $B = n!$ (exactly), you could count the trailing zeroes, as here.
[EDIT]: $f(n)$, approximation of inverse gamma, is defined as $\frac 1 2 + \frac{L(n)}{W(L(n)/e)}$; where $L(x) = \log \frac{x+c}{\sqrt{2\pi}}$, $c = \frac{\sqrt{2\pi}}{e} - \Gamma(k)$.
Here $k$ denotes the positive zero of the digamma function, approximately
$1.461632$. Finally, $W$ is the Lambert function. Source.
For example, starting with $n = 9.3326 \cdot 10^{157}$ and $k \approx 1.4616$ we get $f(n) = 100.9999095692868$ (Mathematica).
