How long can any one service member stay at one base?

There is no set number of years they can keep you on a base. There are a set number of years they have to keep you at a base (stabilization), after that time is up they can move you, but they dont have to. Four years isnt long to be stationed somehwere. I'm at 20 years this year and have only been to three duty stations. I was at Fort Bragg for 11 years prior to my PCS late last year.

Technically, that is incorrect. The army can keep you at a base your entire career. However, they will typically, at a minimum, move you to a different MACOM every few years. An example: As a PSYOPer, I could stay at Fort Bragg, in theory, 20 years, and retire. However, if I did so, I might be in a PSYOP group (sorry - MISO Group now) for 3 years in a BN. Then, they move me out of that group over to the 82nd, USASOC, or the JFKSWC, or somewhere else outside of my normal command. I then stay there for a few years, and then come back to the PSYOP group. All those units are located at Fort Bragg (common knowledge and you can even look it up). I never left Bragg, but I changed MACOMs - technically counting as a "same station PCS". Other MOSes, will typically move every 3 - 4 years, though, I have had Soldiers who were at a base for 6 or 7 years - with a MACOM change.

Call/email your branch manager, that's what they are there for. How long someone stays at any particular duty station has a lot to do with so many different things. If you are ready to leave send an email to your branch manager, wait 72 hours for a response, if you don't hear back, call your branch manager.