I decided to find out if there is a limit and whatever else I could learn. I found two more races that would give me an exploration treaty, so my total went to five. I could not offer an exploration treaty myself because I did not have the tech for it. One of the races that I have an exploration treaty with was willing to trade me the tech. After I got it, I found there was only one race left that I could offer it to, but I was not allowed to do so. The tool tip said "Your instance limit of 1 has been hit" despite not giving any.
I waited a few turns until the first two expired, leaving me with three again. I still could not give an exploration treaty. I guess I can't offer the treaty if an alien race has already given me one. I probably can, however, have as many as there are other races in the game if they are the ones doing the offering. I don't know if they can offer me one if I have already given one to some other race.
I'm not sure if this is the way it is intended to work or not, but that is the way it is.