Let $M$ be a smooth manifold of dimension $p$. Then $TM$ is a smooth manifold of dimension $2p$. Since $TM$ is a smooth manifold it supports a notion of smooth functions on it.
I have never seen this explicitly written (or maybe I didn't understand) but it seems that function on $TM$ should be given by elements of $T^{*}M=Hom(TM, C^{\infty}(M))$.
Is the correct?
Some very nice functions on $TM$ are given by elements of $T^\ast M$, but the vast majority of smooth $f:TM\rightarrow \mathbb{R}$ are not of this form.
For example, if $M = S^1$, then $TM$ is diffeomorphic to $S^1\times \mathbb{R}$. The smooth functions $f = f(\theta, t)$ on $S^1\times \mathbb{R}$ which come from $T^\ast M$ are precisely those where $f(\theta,\cdot):\mathbb{R}\rightarrow \mathbb{R}$ is linear.
Then, for example, the map $f(\theta, t) = t^2$ is not of this form. Nor is something like $f(\theta, t) = \sin(\theta + t)$.