In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as:
If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, $\alpha^\beta$ is the least ordinal greater than any ordinal in the set $\{\alpha^\gamma:\gamma<\beta\}.$
That lead me to think, what is $1^\omega$?
According to the definition above, $$1^\omega=\max\{1^\gamma:\gamma<\omega\}+1=\max\{1^\gamma:\gamma\in\mathbb N\}+1=\max\{1\}+1=1+1=2$$ Is this reasoning correct?
Your reasoning is correct, but Mathworld's definition is wrong: it should specify the least ordinal greater than or equal to all the ordinals in the set $\{ \alpha^\gamma : \gamma < \beta \}$, with the result that $1^\omega = 1$. More generally, $1^\alpha = 1$ for any ordinal $\alpha$.