Much later I encountered the idea in advanced topics in theoretical physics such as random matrix theory and in theoretical chemistry (non-adiabatic transitions and conical intersections).
Yet level repulsion is a very simple phenomena that can be illustrated with just a two by two matrix describing two coupled quantum states, as nicely discussed on the Wikipedia page.
Last semester when I was teaching Solid State Physics I realised just how central and basic the phenomena is and that the students did not appreciate this.
Level repulsion is the origin of several key phenomena in chemistry and physics.
In solid state physics, it is the origin of the appearance of band gaps at the zone boundary and thus the all important distinction between metals and insulators.
Previously, I posted how Chemistry is quantum science because chemical bonding (the lowering of energy due to interacting atoms) arises due to the superposition principle. This could also be viewed as level repulsion.
Another key idea in chemistry is that of transition states and activation energies for chemical reactions. When one uses a diabatic state picture, particularly as emphasised by Shaik and Warshel, the transition state emerges naturally in terms of level repulsion.
The figure is taken from here.
Can you think of any other nice examples?