An example of non-Hamiltonian systems

That's easy. Hamiltonian mechanics describes reversible dynamics. Just introduce irreversibility in your system. like friction, dissipation, viscosity etc.

Can you answer the question now?

We define a Hamiltonian system to be the triad $(H,\mathcal{M},\omega)$ of a Hamiltonian function $H$ on a state-space manifold $\mathcal{M}$ that that is equipped with a (closed) symplectic form $\omega$.

Two long-known and much-studied yet (relatively) simple examples of energy-conserving yet non-Hamiltonian dynamical systems are the (1) the Chaplygin Sleigh, and (2) the rattleback.

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Note Added  In particular, the reason that the dynamics of the Chaplygin Sleigh are not Hamiltonian is geometrically elementary: the state-space manifold of a Chaplygin Sleigh is odd-dimensional — namely, the x and y spatial coordinates of the sleigh, the angular orientation of the sleigh, its linear momentum, and its angular momentum — whereas symplectic forms exist only on even-dimensional manifolds.

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Viewed as a flow on $\mathcal{M}$, the dynamics of these systems is energy-preserving but not a symplectomorphism. In thermodynamic terms, the First Law holds, but the Second Law need not.

For example, in we read in Advances in the Theory of Control, Signals and Systems with Physical Modeling:

One of the striking features of non-holonomic systems is that while they conserve energy they need not conserve volume in the state space.

The study of the thermodynamic properties of ensembles of these systems (and other non-symplectomorphic systems like them), and their quantum generalizations, are active areas of research.