are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?
Actually the first case in history of a symplectic manifold wasn't a cotangent space. It was the space of Keplerian motions of a planet, represented locally by its Keplerian elements. The Lagrange symplectic structure on this space is defined by the so-called Lagrange parenthesis he introduced at this time (three papers in 1808/09/10)(*). That manifold is actually even non Hausdorff, but its greatest hausdorff quotient is a still a manifold (this is known as the "regularization theorem"). This manifold is symplectic but not a cotangent (but however contractible to the sphere $S^3$). Extended with the repulsive motions, it is an algebraic manifold defined by the following equations [Sou]. $$ \left\{ \begin{array}{rcl} ||{A}||^2 -fx^2 & = & 1 \\ y^2 - f ||{B}||^2 & = & 1 \end{array} \right. \quad \& \quad \left\{ \begin{array}{rcl} A \cdot B - xy &=& 0 \\ xy - f\tau &=& 0, \end{array} \right. $$ where $A,B \in {\bf R}^3$ and $x,y,f,\tau$ belong to $\bf R$. Actually this manifold is the result of the gluing of $TS^3$ and $TH^3$ along $TS^2\times {\bf R}$ (where $H^3$ is the 3 dimensional pseudo-sphere). I made the following picture, for $A$ and $B$ in $\bf R$, to get a visual idea of the manifold. The bottom represents the $TS^3$ part, the top represents $TH^3$ and it is glued along two lines representing $TS^2\times {\bf R} \simeq S^2 \times {\bf R}^3$.
Remark. There exists also the examples of compact symplectic manifolds representing internal degrees of symmetries, as mentioned in Tobias answer. In the same spirit there is the Grassmannian manifolds ${\rm Gr}(2,n+1)$ of $2$-planes in ${\bf R}^{n+1}$, representing the space of un-parametrized geodesics on the sphere $S^n$. We can regard this space of geodesics as the space of light rays on the Euclidean sphere where the speed of light would be infinite.
---------- Edit March 28, 2017
On a conceptual point of view, I just finished to write a paper:
Universal Structure Of Symplectic Manifolds
http://math.huji.ac.il/~piz/documents/ESMIACO.pdf,
---------- Edit November 15, 2019
http://math.huji.ac.il/~piz/documents/ESMIALCO.pdf,
This paper has been enhanced to make the symplectic manifold an orbit of the linear coadjoint action of a central extension of the group of Hamiltonian diffeomorphisms, independently of the group of periods. That is, even if the symplectic form is not integral.
that proposes a way, based on diffeology, to understand the statement: "Every symplectic manifold is a coadjoint orbit".
---------- Notes
(*) I published this paper on the origins of symplectic geometry, but in french, where Lagrange's construction is explained.
---------- Reference
[Sou] Jean-Marie Souriau. Géométrie globale du problème à deux corps. In Modern Developments in Analytical Mechanics, pp. 369-418. Accademia della Scienza di Torino, 1983.
The classical phase space of the spin degrees of freedom is represented by the two-sphere with a symplectic form given by $s$ times the standard volume form. This is clearly no cotangent bundle to some configuration space (and in particular non-exact). Geometric quantization results in the well-known quantization $2s \in \mathbb{Z}$.
By the way, examples arising from symplectic reduction are very natural and important. Often this is the only procedure available to endow a space with a symplectic structure.
I think from the physical point of view, the question should be reversed. Symplectic and presymplectic manifolds naturally occur in physics because such geometric structures naturally follow from a variational formulation of the equations of motion of a mechanical system or field theory (see here). This variational character is indeed connected with quantization, but there is no need to go into that here. So the interesting question should be: when are symplectic manifolds occurring in physics actually cotangent bundles?
The answer is that happens naturally only if fairly special cases. Namely, the phase space (space of initial data with symplectic structure) of a mechanical system (some field theories can be considered as infinite dimensional mechanical systems) is a naturally a cotangent bundle when the system's equations of motion are second order in time, the kinetic term is non-degenerate and non-holonomic constraints are absent. I'm being a bit vague with the terminology, so let me briefly expand on these conditions.
If $Q$ is the configuration manifold, then the natural space of initial data for a $(k+1)$-st order ODE on $Q$ is the $k$-jet bundle $J^k(Q\times \mathbb{R})_{t=0}$, where $Q\times \mathbb{R} \to \mathbb{R}$ is the trivial bundle over $\mathbb{R}$ (time $t$) with fiber $Q$. It just so happens that, given a Lagrangian defined on $J^1(Q\times\mathbb{R})$, so that the equations of motion are second order, if the Legendre transform is well defined, it establishes an isomorphism $J^1(Q\times \mathbb{R})_{t=0} \cong TQ \cong T^*Q$. In other cases, it might be possible enlarge the configuration manifold to $Q'$ and then through a sequence of transformations on the equations of motion to show that $J^k(Q\times\mathbb{R})_{t=0} \cong T^* Q'$, but I would not necessarily call that identification natural.
If the kinetic term of the Lagrangian is singular (there is no invertible map between the canonical momenta and the velocities), then the Legendre transform is not well defined and, even in the $k=1$ case, the naive identification from the preceding paragraph fails. A cotangent bundle can make an appearance here as well, but only with constraints. That is, it is possible to extend the configuration space to $Q'$ and then ultimately the space of initial data can be identified with a quotient of a submanifold of $T^*Q'$ (submanifold defined by second and first class constraints, while the quotient is defined by the flow generated by first class constraints). So, while a cotangent bundle $T^*Q'$ is used in this construction, I would not say that the end result is naturally a cotangent bundle itself.
Finally, there are special situations where the constraints are holonomic. That is, it is possible to satisfy all constraints by simply restricting the mechanical system to a submanifold $Q''\subset Q$ of the original configuration manifold. Then, for second order equations with a well defined Legendre transform, the space of initial data is once again a cotangent bundle, $T^*Q''$. However, if the constraints are more complicated, that is, non-holonomic, the identification with a cotangent bundle once again fails.
So, as you can see cotangent bundles appear as phase spaces in physics only under special conditions. It so happens that there are many examples of simple mechanical systems that satisfy these conditions, but one does not need to go far to find examples that do not.