I think $SL_2(\Bbb R)$ an example. $ A = \begin{pmatrix}
-1 & 1 \\
0 & -1 \\
\end{pmatrix}$ is not the matrix exponential of a real $2\times2$-matrix. Suppose $A = \operatorname{exp}(B)$, with $\operatorname{tr}(B)=0$. If $B$ is nilpotent, then $B$ is similar to $0$ or $\begin{pmatrix}
0 & 1 \\
0 & 0 \\
\end{pmatrix}$, but the matrix exponential of that has the wrong eigenvalues. So if $B$ is not nilpotent $\operatorname{tr}(B)=0$ implies that $B$ is diagonalizable, but the matrix exponential of a diagonalizable matrix is diagonalizble, but $A$ is not diagonalizible either.