I am fairly certain that all three keys did not come from the Doctor. By this point, Martha Jones had her own key...
So that's one from the Doctor and one from Martha Jones.
And I believe I recall Jack fumbling to get his off a key ring.
So the answer isn't that the Doctor carries 3 keys or more.
It's that he carries ONE.
But as to how many copies he has, that's at least a little established in the episode where Clara steals them all and destroys them all, in a shared dream sequence. By all, I mean all that Clara knows about, which means there are likely a few more. Of course by this time, he just needs to snap to get in, but I guess that she doesn't know that...I can't find clips for either of these but the episode Dark Water and The Sound of Drums will show you what you want at least a little.
It is never actually established as to how many exactly he has made, or how many he generally carries. I am thinking that it has to be ONE as far as carries, because it's a very valuable thing to be carrying around. And again, when he's lost it, it's been a big deal.