A planar graph cannot have $K_5$ and $K_{3,3}$ as minors. RobertsonSeymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are obstructions that prevent the graph from being genus $g$. Is there any result on the size of the list? Is it linear in $g$?
No, it is not linear in the genus; it is at least exponential in $g$. See for example this answer by David Eppstein.

$\begingroup$ i dont see a reference for 'at least'. $\endgroup$ – user76479 Jan 14 '16 at 14:21

1$\begingroup$ As Eppstein mentions, if you take a graph which consists of exactly $g+1$ blocks, each of which is a $K_5$ or a $K_{3,3}$, then this will be an excluded minor for embedding in a surface of genus $g$. There are obviously at least $2^{g+1}$ such graphs. $\endgroup$ – Tony Huynh Jan 14 '16 at 15:43
It seems that this paper by Djidjev and Reif establishes an upper bound of $\exp(O(g)!)$ for the number of minimal forbidden minors.
Djidjev, Hristo, and John Reif. "An efficient algorithm for the genus problem with explicit construction of forbidden subgraphs." Proceedings of the twentythird annual ACM symposium on Theory of computing. ACM, 1991.