Chaining 25 74HC595 shift registers together
In the ideal case, driving 25 HC595s will work just fine.

However, there are some things you can do to make it not ideal, especially easy if you have long lines or high current consumption.

If you have a very long clock and strobe lines, then reflections could be a problem and you need to treat them as transmission lines. Usually a series resistor at the source end of a single, long, unterminated line is sufficient.

If you have your power ground and logic ground as the same conductor, then current flowing in a long line could generate enough V=IR voltage drop to shift the ground voltage at the far end such that you don't make the logic levels for clock and strobe signals send from the control end. You might want to rebuffer the clock and strobe at each board, if you have a long string of boards, so the voltage shift doesn't build up.