Radix sort and changing bases
The asymptotic runtime ignores access times for the digits.  On typical hardware a base 256 number can be manipulated faster than extracting base 10 digits from the same number.

If you have multiple CPUs, you can more easily exploit parallel processing by using a larger base.  The large base will spread the numbers across many buckets which can then be sorted in parallel.

Another reason to change bases is if you know your data has a bias (such as Benford's Law) that would cause the numbers to initially cluster into only a few buckets.