Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$ If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for different volatilities? Is there a "tipping" point where its limiting distribution behavior changes?

Just to be clear, I'm assuming that you mean the process $dX = \sigma X dW$ where $W$ is Brownian motion.  According to the theorem 12 from these lecture notes at Brown University, driftless geometric Brownian motion converges to zero with probability 1.  They sketch a proof in terms of the law of the iterated logarithm.
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Loosely related is this answer to a different question, which makes an interesting big-picture point.  At any finite time, $E_0(X_t) = X_0$, so its surprising that the long-run limit is zero.  The answer locates this weird result in a failure of a probabilistic property known as "uniform integrability".
