What is the oldest you would be permitted to join the military?

Basically, as old as you can be while being able to serve 20 years.
Of course, each service establishes their own business rules, guidelines, and exceptions based on need

Depends on what your specialty is and how bad you’re needed. I went to DCC recently and there was surgeon in the course who was brand new to the army and came in as a LTC with no experience. He was in his 60’s and had been in his specialty for decades. They needed him and he was waivered in.

As others have stated the age limits change for each service quite a bit. For example when I enlisted in 1996 I was 27 and was told by Marines, Navy and Air Force that I had to enlist by my 28th birthday and the Army limit was 32, but again changes often so best bet is to call recruiting offices and ask. Good luck