If we work solely from the standard mathematical formula for Elo ratings, it's possible to prove that the winning player can get an arbitrarily high rating, but that it'll probably take exponentially long (i.e., the winning player's rating increases logarithmically).
I'll suppose that the initial Elo rating of the two players is 0, not 1500, because it makes the math simpler. (If this bugs you, just add 1500 to every number below.) Let r(n) be the rating of the winning player after n games, so the rating of the losing player is -r(n). Then in the (n+1)st game, the rating difference is 2r(n).
By the calculation given in Brian's answer, r(n) obeys the formula
This recursion is painful to solve explicitly, but we can use a differential equation to approximate the solution. Explicitly, let f(r)=10^(r/200)+1; then the above recursion can be approximated as
Intuitively, passing to the differential equation is considering the limit of the process where we repeatedly play twice as many games, but with a K-factor that's half as large. Every time we do this, we slow down the rating growth a little bit, because the winning player has a higher rating as they play the second game in each pair than they do in the first game, so the second game leads to slightly less Elo growth than the first game.
The differential equation r'(t)=K/f(r(t)) is a separable differential equation, which can be solved. I won't bore you with the details on a platform that doesn't let me write formulas, but the solution turns out to obey the equation
t=(200e^(r/200)-200+r ln 10)/(2K ln 10) .
Notice that this is an equation for t (the number of games played) in terms of r (the rating of the better player). There's essentially no nice way to invert this equation to write r in terms of t. However, we can notice that t is well-defined for any positive r, and is an increasing function of r. So, for any given rating, we can find the amount of time required to achieve that rating, which means it's possible to achieve arbitrarily high ratings.
But since t is an exponential function of r, the amount of time taken may be exponentially long; that is, r will grow logarithmically over time.
This argument shows that the winning player's rating can grow arbitrarily large in the limit where we played many more games, but with a correspondingly smaller K-factor. But as we argued above, the winning player's rating increases faster in the original setup than it does in this limiting setup. Since ratings can grow arbitrarily large in the limiting setup, this means that ratings can also grow arbitrarily large in the original setup.
I'm reasonably certain (but have not proved) that this growth is still logarithmic.