The right-most non-zero digit

It is a common exercise to determine the number of trailing zeros of a factorial. For example, $20! = 2\; 432\; 902\; 008\; 176\; 64{\color{#20A900}{0}}\; {\color{#20A900}{000}}$ has 4 trailing zeros (as highlighted in green above).

However, finding the rightmost non-zero digit of a factorial is much harder. For example, the rightmost non-zero digit of $20!$ is 4, as shown above.

Problem: Find the rightmost non-zero digit of $10000!$.