What is the maximum integer that can be saved in a LaTeX counter?

The maximum value is the 'usual' 2^31-1 long integer value, as well as the negative range from -2^31, so the full range is 2^32 integers possible.

2^31-1 is 2147483647, which is the largest possible integer usable for counters or \ifnum and \numexpr codes.

In the code below I stored this number to the counter \mycounter and print it several times after using \stepcounter. After the first \stepcounter the register overflows and the number is set to -2147483648, being the 'largest' negative number possible. A subsequent \stepcounter works normally then.

The e-TeX standard extended the limit of 256 registers (count, skip etc.) to 32568 possible registers (for each type)

\documentclass{article}

\newcounter{mycounter}

\begin{document}

\setcounter{mycounter}{2147483647}

\themycounter  % prints 2147483647 

\stepcounter{mycounter}  % Now the overflow will occur

\themycounter % prints -2147483648

\stepcounter{mycounter}

\themycounter % -2147483647
\end{document}

According to The TEXbook:

TEX has 256 registers called \count0 to \count255, each capable of containing integers between -2147483647 and +2147483647, inclusive; i.e., the magnitudes should be less than 2³¹.