Why can free lithium atoms not take part in an Auger process? Shouldn't it be possible for an incoming photon to excite one of the 1s electrons to a 2p state (or one of even higher energy) and then for the excited electron to drop back to 1s and kick out the 2s electron?
Is there some fundamental reason that prohibits this process, or is it just very unlikely, or is it not called "Auger process" because the first electron is excited instead of kicked out?
 A: I am no expert on the Auger effect but the wikipedia page on Auger electron spectroscopy seems to indicate that Auger effects can be observed in lithium. As @WillieWong points out in the comments, the usual Auger effect does not work: if the laser simply removes one 1s electron, you are left with a 1s$^1$ 2s$^1$ configuration, which cannot ionize to Li$^{++}$ as the excited electron can only bump the ground one up to 2s (and even that transition is out since the final state is identical to the initial one!).
It is in principle possible, however, for the laser to simultaneously - or sequentially - ionize a lithium atom and excite the ion, leaving it in a configuration 2s$^1$ 2p$^1$ or higher. (As said in the comments, 2s$^2$ is out by dipole selection rules.) Such a configuration (or a similar one) would have enough energy to expel one of the electrons in an Auger transition.
The catch is that while a single laser pulse can do both things, a single laser photon can't. Leaving a doubly-excited ion is, for weak pulses, a two-photon process, since you need to do two transitions, each of which annihilates a photon. As such, it is twice as unlikely as the 1s$^1$ 2s$^1$ transition, or to put it another way it scales quadratically with the laser intensity. Fortunately, it is possible to produce pulses strong enough to use this to their advantage (or indeed to take the situation far out of the perturbation regime where photons are meaningful concepts) and use strong intensities to leave all sorts of interestingly excited ion electronic configurations. As long as some of these result in Auger electrons, these will be relatively easy to detect as they have quite characteristic signatures.
