In addition to all the answers that are correct in a mathematical sense, in a practical sense, a system whose frequency response goes below some finite but small-enough value will not be usefully invertable, even if a simple mathematical analysis would suggest that it is.
In frequency-domain terms, the frequency response of a system's inverse will have gain ...
Whether LTI or not all systems are invertible if
unique (distinct) inputs produce unique (distinct) outputs
Causality and stability are later concerns for making sense of the obtained inverse system.
For example the inverse to the delay system
$$y[n] = x[n-d] $$
$$y[n] = x[n+d] $$
Which is clearly noncausal for $d > 0$, and is not ...
[EDIT] A necessary condition for invertibility is that any output has only one possible input (or injectivity, as proposed in comments). Since we are looking at counterexamples, we can look at when this condition is not satisfied.
The null system, that turns every signal into a zero flat line, is not invertible, but a bit trivial.
A system that computes a ...
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 $y(t)=x(t-T)$, $T>0$, could only be ...