Hmm… if you count smallish aid missions (the kind that can hardly be more than aid or peacekeeping missions due to their size) you can run things quite high. After all, virtually every full-rank US Embassy has a Marine detachment on guard. A news release from the US Army in 1997 showed nearly 100 countries had Army personnel on some sort of mission, from two officers on short-time UN duty in Guatemala to tens of thousands permanently in South Korea and Europe.
Here’s one list of “major” US military bases
These refer only to major permanent bases. So it does not mention Afghanistan, Georgia, Uzbekistan, Camp Bondsteel in Kosovo, Phillipine or Colombian joint missions, Laos/Cambodia mineclearing ops, Kuwait, etc.
Here is one critic’s article that includes his numbers on the question It says 25 countries have “significant deployments” and the total adds up to around 200,000 on the ground.