Answer: All $C\leq \frac{3^3}{4^4}$ To see that these constants work set the sequence to $a_i=\frac{3}{4}$. Now assume $C>\frac{3^3}{4^4}$. It suffices to only look at positive sequences. Notice that for all positive $x$ we have that $x^3<x^4+C$. Thus we have $a_{n+2}\leq \max(a_{n},a_{n+1})$. Let $M_k=\max(a_{2k-1},a_{2k})$. Then $$M_k^3\geq M_{k+1}^4+C$$Let $C=\frac{3^3}{4^4}+\epsilon$. Then $$M_k^3\geq M_{k+1}^4+\frac{3^3}{4^4}+\epsilon \geq M_{k+1}^3+\epsilon$$Thus $M_i^3$ will eventually become negative, a contradiction.