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?)
 A: There is more to the picture than just the superposition principle applied to long state vectors. @Lubos notes that This number of basic configurations is exponentially large because the spins are independent, right. But there are certain sectors of the Hilbert space which are distinguished from others in the amount of entanglement among the constituent spins of state vectors in the respective sectors.
For example if I have two spins, then I can have states such as: $(|0\rangle|1\rangle + |0\rangle|0\rangle)/\sqrt{2} = |0\rangle(|1\rangle+|0\rangle)/\sqrt{2}$ which are factorizable and those which are not such as $(|0\rangle|1\rangle \pm |1\rangle |0\rangle)/\sqrt{2}$. This is the simplest example of an entangled state and is called bipartite entanglement. In general one can have a state with entanglement among more than 2-spins.
For a state with $n$ spins as $n$ grows so does the dimension and complexity of the subspace describing sequences with bipartite, tripartite and so on to k-partite entanglement ($k\le n$) among the constituent spins.
Apart from the possible ways to exploit multipartite entanglement for quantum computation, there is the question that @Jim asks:

Why don't we observe multipartite entanglement in the real world?

Of course one response is simply that such states are very fragile and decay incredibly fast into non-entangled states due to decoherence and are thus hard to make and harder to observe. Though one might see them in microscopic systems they have no utility for describing phenomena at larger scales.
The question as to whether such states are more stable than this naive picture suggest is still wide open. It is possible, and I strongly believe, that multipartite entanglement will turn out to play a far more important role in our daily lives than we realize at present.
One very compelling prospect is that life, which has been evolving for billions of years and which has developed sophisticated mechanisms which we are only now discovering, has evolved methods to exploit quantum information processing. Here I will just list some references:


*

*Apoorva Patel has been working on the relation between http://arxiv.org/find/quant-ph/1/au:+Patel_A/0/1/0/all/0/1

*Seth Lloyd summarizes these developments beautifully in this video.
Again, neither of these are explicit demonstrations of multipartite entanglement in action but it would be a shame if life failed to exploit such a wonderful resource.

Edit: Let me answer the question with a concrete example of multipartite entanglement in action in the "real-world":
There are certain insulating magnets whose magnetic susceptibility as a function of temperature scales not according to the $1/T$ Curie law, but according to a power law $T^{-\alpha}$ ($\alpha \lt 1$). Ghosh et al, demonstrate with numerical simulations that such power law behavior can be explained only is long-range entanglement of dipoles in the spin-system is taken into account.
To quote from the concluding section of "Entangled Quantum States of Magnetic Dipoles" by Ghosh et al. (Nature, 2003):

There is a growing realization that entanglement is a useful concept for understanding quantum magnets, thus unifying two rapidly evolving areas, quantum information theory and quantum magnetism. The discussions to date have focused on one-dimensional magnets and measures of entanglement with clear theoretical meaning but no simple experimental implementation. Our experiments and simulations represent a dramatic illustration of how entanglement, rather than energy level redistribution, can contribute significantly to the simplest of observables – the bulk susceptibility – in an easily stated model problem. (emph. mine)

A: This exponential growth is what allows us to have multipartite entanglement, and exponentially many Everett branches in the first place.
If in the near future, we can construct a quantum computer implementing Shor's algorithm, this will be a good observational example.
