Because

• $8888\equiv 8\pmod{37}$
• $7777\equiv 7\pmod{37}$
• $8^2 \equiv -10\pmod{37}$
• $7^3\equiv 10\pmod{37}$

$$\therefore 8888^{2222} + 7777^{3333} \equiv 8^{2222}  +7^{3333} \equiv (-10)^{1111} + 10^{1111}\equiv 0 \pmod{37}$$

This means the answer is $\boxed{0}$.