Quantum Mechanical Hamiltonians without Classical Analogues

Recently I found myself in a state similar to that which @senator found himself here. I too have been reading Dirac's Lectures on Physics and am particularly confused by the notion of Hamiltonians without classical analogues.

The way I understand it, at Second Quantisation one always starts with the classical Lagrangian to end up with the Quantum Mechanical Lagrangian and so within this capacity I do not see how each Hamiltonian cannot have a classical analogue.

Is this still the route taken to obtaining Hamiltonians which do not have classical analogues and if so what are some examples of this?

If it is the case that at each instance of finding a Hamiltonian one starts from a classical Lagrangian, is that not wrong given that Classical Mechanics is a subset of Quantum Mechanics.