Are all LTI systems invertible? If not, what is a good counterexample?
You need to define what you mean by "invertible". Do you mean invertible by a causal and stable system? If yes, then any system that is not minimum-phase is not invertible (because the inverse system can't be causal and stable).

Example of a system that cannot be inverted by a causal and stable system: a simple delay $y(t)=x(t-T)$, $T&gt;0$, could only be inverted by a non-causal system.

If by "invertible" you mean a system that can be inverted by a system that is possibly not causal and/or not stable, then it's still straightforward to find systems that can't be inverted according to this broader criterion: just take a system with one or more zeros in its frequency response. The information about the input signal at the frequencies where the system's frequency response is zero cannot be recovered by any other system. 

Example of a system that cannot be inverted by any system: an ideal low-pass filter cannot be inverted by any system, because any information above the filter's cut-off frequency is lost and cannot be recovered.

EDIT: the latter definition of invertibility (where causality and stability of the inverse system are ignored) is just the definition of an injective mapping. In terms of systems this means that no two distinct input signals generate the same output signal. This category is what is also described in Laurent Duval's answer. Note, however, that in practice we usually prefer inverse systems that can also be realized, i.e., we require stability and causality, and then the first definition of invertibility given above is appropriate.