Complexity of dynamic card game algorithm
As pointed out in the comments, your recurrence is wrong (though equivalent for $b=r$; it's a recurrence for $E(b,r)+b-r$). Also, you can solve this efficiently using memoization or its more principled cousin, dynamic programming. The dynamic programming solution calculates iteratively $E(i,j)$ for $i+j=0,1,\ldots,52$.

Finally, to answer your question, if you implement you original solution, you get a running time satisfying the recurrence
$$
T(b,r) = C + T(b-1,r) + T(b,r-1),
$$
with initial conditions $T(b,0) = O(1)$, $T(0,r) = O(1)$. If $T'(b,r) = T(b,r)-C$ then
$$
T'(b,r) = T'(b-1,r) + T'(b,r-1),
$$
a recurrence solved by $\alpha \binom{b+r}{r}$. Taking the initial conditions into account, we get that
$$
T(b,r) = \Theta\left(\binom{b+r}{r}\right).
$$
In particular,
$$
T(n,n) = \Theta\left(\binom{2n}{n}\right) = \Theta\left(\frac{4^n}{\sqrt{n}}\right).
$$
(This assumes all arithmetic is $O(1)$. The running time is actually somewhat larger since the relevant numbers grow fast, but this is a (multiplicative) lower-order factor.)