Has every possible interaction between elementary particles been observed?

• proton decay is un-observed, and suspected on the basis of various Beyond the Standard Model theories
• Nothing directly involving the Higgs had been published when the question was asked, but ATLAS and CMS have pretty definitive observations of the most easily measured channels. Some of the harder channels are still under invesitgation.
• Neutrinoless double beta decay would be the signature of $\nu + \nu \to \text{::nothing::}$ (off-shell, of course) and would indicate that neutrinos are Majorana particles. There is a report of it, but the significance is limit and it is unconfirmed
• Evidence for any kind of dark matter interactions outside of gravity is pretty sparse on the ground
• No super-symmetric partners have been observed
• I doubt anyone has seen $\nu_\tau + n \to \tau + p$, though this is required by the current electoweak formalism. Or at least, they haven't been able to show that this is what they saw

Basically lots of dark corners. Note that much of this is Beyond the Standard Model, so may or may not represent the real state of physics.

dmckee has given a good, thorough answer. I'd bet that there are lots of other interactions that haven't been seen. How about $\gamma+\gamma\to Z+\overline Z$, or $\gamma+\gamma\to t\overline t$? To see those reactions, you'd need high-energy photons, and the cross sections should be very low. I doubt that we've produced the right experimental setup for those reactions. Perhaps even better would be a ton of similar $\nu\overline\nu$ reactions: $\nu_\tau+\overline\nu_\tau\to t\overline t$, for instance.

All of these satisfy the conservation laws and are predicted to have nonzero cross sections in the standard model, but I'd be surprised if they've been observed.

There are some extremely important reactions that have never been observed directly. My favorite example is p-p fusion,

$$\text{p} + \text{p} \rightarrow \text{d} + \text{e}^+ + \nu_e$$

which is the rate-determining step for the main fusion process in the Sun and all other small stars. But it is utterly impossible to observe this reaction in anything like today's accelerators, because the cross section is so tiny. It's only important in the Sun because there's such a large, dense collection of hot protons. (In fact the smallness of this cross section is the reason all the stars haven't burnt out yet.)

Other examples would be any reactions of two photons, or two neutrinos, as Ted Bunn said.