1. The problem is to evaluate the expression $e^{\infty^2}$. 2. Recall that $\infty$ represents an unbounded quantity that grows without limit. 3. Squaring $\infty$ still results in $\infty$, since $\infty^2 = \infty$. 4. Therefore, the expression simplifies to $e^{\infty}$. 5. Since $e^{\infty}$ means $e$ raised to an infinitely large power, the value tends to infinity. 6. Hence, $e^{\infty^2} = e^{\infty} = \infty$. 7. In conclusion, the expression diverges to infinity and does not have a finite value.