Zor gets very close, but I still think he’s wrong:
First, since people seem to think it’s an important issue, I’ll say that I don’t know what percentage of the mixture in the balloon is helium, but I also don’t think it matters. All we care about is that it’s some mystery mixture that’s lighter than air, and some or all of its components can leak out slowly, leaving a mystery mixture whose makeup is also lighter than air, but possibly less so.
The “air entering the balloon as the helium escapes” business seems patently bogus to me. If you consider that the mystery mixture inside is at a higher pressure than the air outside, the porous balloon wall is only going to pass gas (har) in one direction: out.
That said, the behavior is still easily explainable. When the balloon is brand new, the combination of the balloon itself and the mystery mixture inside weighs less than the corresponding volume of air, so it floats. As the mystery mixture leaks out, the balloon+gas system loses weight, as opposed to gaining weight as you say. However, the size decreases rapidly enough that with the weight of the balloon material itself remaining constant, the density of the balloon+gas system eventually surpasses the density of air, and it sinks.
How small the balloon can get and still float is basically a function of the weight of the balloon material (I say basically because there is still a question as to whether the makeup of the gas mixture inside the balloon changes as it leaks, but I bet that’s not nearly as large an effect as others are making it out to be). If you could eliminate the heavy knot, you’d probably significantly increase the life of your balloon.
Note that this explanation still works if we believe that air actually seeps into the balloon (which I personally think is ridiculous), because the density of the whole system will increase even more rapidly.
As to why, if the mystery mixture can leak out, does the balloon not go completely flat, I think jayron32 hit it right on the head. In layman’s terms, it leaks out easier when it’s under more pressure, and as it comes closer and closer to atmospheric pressure, it leaks out more slowly and probably stops somewhere slightly above 1 ATM. In my head, I have a picture of a stretched out piece of latex whose holes get smaller as it unstretches, eventually getting to the point where the holes are too small.