The obvious answer is brute force, thus...
while (a_iplus1 < 10) && (iter < imax)
[a_i,a_iplus1] = deal(a_iplus1,a_iplus1 + 1/a_i);
So when iter = 48, a finally grows larger than 10.
Note that I wrote the code so that no arrays are grown. This is important, since the loop might have gone on forever. This is why I put an upper limit on the loop. The trick using deal to advance the terms is well, just a cute trick.
Does a general analytical solution to this nonlinear difference equation exist? Possibly. Such nonlinear difference equations tend to have "interesting" behaviour. As soon as you dive into the domain of the nonlinear, things can go straight to well, you know where. The plot however, suggests a simple asymptotic behavior, one that makes sense when you look at the expression. Questions now come up, like is there a limiting value? Or will a(i) grow forever, unbounded?