So I understand that a chaotic system is a deterministic system, which produces aperiodic long-term behaviour and is hyper-sensitive to initial conditions.
So are all aperiodic systems chaotic? Are there counter-examples?
A system might be, for instance, stochastic, random - which is certainly not an example of deterministic chaos, but is aperiodic.
You can also have quasiperiodic behavior, where the system comes close (but not exactly) to previous states, for which the simplest example is the circle map: $x \mapsto a+x$, where $a$ is irrational and $x$ is angle-like (e.g., $x\mapsto x\in[0,1)$). This dynamics is aperiodic, but not sensitive to initial conditions.
Also a damped harmonic oscillator doesn't come back to exactly the same state, since energy is being lost and, thus, is strictly not periodic (nor sensitive to initial conditions).
These deterministic aperiodic but non-chaotic behaviors are often called regular, as in the classic book Regular and Chaotic Dynamics by Lichtenberg and Lieberman.
A related question is Conditions for periodic motions in classical mechanics.