quantum mechanics - Observing the exponential growth of Hilbert space?

One of the weirdest things about quantum mechanics (QM) is the exponential growth of the dimensions of Hilbert space with increasing number of particles. This was already discussed by Born and Schrodinger but here's a recent reference:

http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.4770v2.pdf

and another one:

http://www.scottaaronson.com/papers/are.pdf

Another difference between quantum mechanics and classical physics is that QM is discrete.
A good example is the line spectrum of hydrogen.

Is there any good observational example which demonstrates the exponential growth of Hilbert space dimensions?
Even for low numbers?
(Does it perhaps show up in the line spectrum of a three or four valent atom?)