Manifolds with polynomial transition maps Title edited I  thank  მამუკა ჯიბლაძე  and  Corbennick for  their  suggestion on the  title  of this  question. I  changed the  title based on the suggestion of  Corbennick.
What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:
For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its  components are  polynomial  functions).
Motivation: The  motivation for this  question comes  from the  concept "Affine  manifolds". I  am  indebted to Mike  Cocos,  for  learning this concept and its  related problem, that is  Chern  conjecture.
 A: Let me develop Robert Bryant's answer :
if $M$ is a simply-connected manifold of dimension $n$ with a polynomial atlas, then there exists a local isomorphism $M \rightarrow \mathbb{R}^n$. Conversely, such a local isomorphism yields a polynomial atlas.
Indeed let $\mathcal{F} \subseteq \mathcal{C}^{\infty}_M$ be the subsheaf of polynomial functions (with respect to the given atlas). Then the pair $(M,\mathcal{F})$ is locally isomorphis to an open subset $U$ of $\mathbb{R}^n$ together with the constant sheaf associated to $\mathbb{R}[X_1,\dots,X_n]$. Thus $\mathcal{F}$ is a local system.
Since $M$ is simply-connected, the locally constant sheaf of $\mathbb{R}$-algebras $\mathcal{F}$ is constant. Thus there is an isomorphism $\mathbb{R}[X_1,\dots,X_n]_M \simeq \mathcal{F}$. The images of $X_1,\dots,X_n$ are smooth functions on $M$,  which give a local isomorphism $M \rightarrow \mathbb{R}^n$ (compatible with the polynomial atlas).
Note that the map $M \rightarrow \mathbb{R}^n$ has open non-empty image, hence non-compact image if $n >0$. Thus such a $M$ is never compact for $n>0$.
A: If I remember correctly, this is impossible for any (nonempty) simply-connected compact manifold of positive dimension.  In particular, $S^2$ cannot have such an atlas.  Off the top of my head, I don't remember where I saw this statement, but I'll try to find a reference.
Followup:  I still don't remember the reference, but I think that the proof goes something like this:
By hypothesis, both $\tau=\psi\circ\phi^{-1}$ and $\sigma = \phi\circ\psi^{-1}$ are polynomial maps, and hence they are defined everywhere on $\mathbb{R}^n$.  By hypothesis there is a nonempty open domain $D_1\subset\mathbb{R}^n$ such that $\sigma(\tau(x)) = x$ for all $x\in D_1$ and a nonempty open domain $D_2\subset\mathbb{R}^n$ such that $\tau(\sigma(y)) = y$ for all $y\in D_2$.  Since $\sigma$ and $\tau$ are polynomial mappings, it follows that $\sigma(\tau(x)) = x$ for all $x\in\mathbb{R}^n$ and $\tau(\sigma(y)) = y$ for all $y\in\mathbb{R}^n$.  Thus, $\sigma$ and $\tau$ are globally invertible.  Moreover, by the chain rule, we have $\sigma'(\tau(x))\tau'(x) = I$ for all $x\in \mathbb{R}^n$, and, taking determinants, it follows that $\det(\tau'(x))$ and $\det(\sigma'(\tau(x)))$, which are polynomial in $x$ must actually be (nonzero) constants.  
Consequently, it follows that $\Lambda^n(T^*M)$ carries a (unique) flat connection such that, if $\mathrm{d}x$ is the standard volume form on $\mathbb{R}^n$, then $\psi^*(\mathrm{d}x)$ is a (local) parallel section of $\Lambda^n(T^*M)$ for each $(\psi,U)$ in the atlas $\mathcal{A}$.  By simple-connectivity, there is a global parallel volume form $\mu$ on $M$, and we can, composing the members of our atlas with linear transformations of $\mathbb{R}^n$, get a new atlas with polynomial transitions that satisfies $\psi^*(\mathrm{d}x) = \mu$ on $U$ for all $(\psi,U)$ in the new atlas.
There is also a way to analytically continue the map $\psi$ coming from a chart $(\psi,U)$ to the domain $U\cup V$ for any chart $(\phi,V)$ for which $U\cap V$ is non-empty:  Since $\tau$ is defined on all of $\mathbb{R}^n$, and since $\psi = \tau\circ\phi$ on $U\cap V$, one can take $\psi(q) = \tau(\phi(q))$ for all $q\in V$, which extends $\psi$ to $V$. 
Using this construction, any $(\psi,U)\in\mathcal{A}$ can be uniquely analytically continued along any smooth path in $M$, and one easily verifies that this extension depends only on the homotopy class of the path with fixed endpoints.  Since $M$ is simply connected, this means that $\psi$ can be extended uniquely analytically to all of $M$, and hence $\psi:M\to\mathbb{R}^n$ is a smooth map that satisfies $\psi^*(\mathrm{d}x) = \mu$ on all of $M$.  Obviously, this is impossible.
A: Brief sketch of slight simplification of Bryant's answer:
Without loss of generality $\mathcal A$ is maximal with respect to the condition that all transition functions are polynomial. Now make a space $\tilde M$ by gluing together $U$ and $V$ whenever $(\phi, U)$ and $(\psi,V)$ are charts such that $\phi$ and $\psi$ agree in $U\cap V$. This is a smooth manifold (or rather each of its connected components is), because polynomials are so rigid. It is equipped with both a covering projection to $M$ and an immersion in $\mathbb R^n$. So if $M$ is simply connected then $M$ immerses in $\mathbb R^n$. If $M$ is also compact and $n>0$ then this is impossible. 
