Can transcendent degree of transcendent number be algebraic? (meaning can both $\alpha^{\beta}$ and $\beta^{\alpha}$ be algebraic)

Question

Here is my question. Are there $\alpha$ and $\beta$ transcendental numbers such that both $\alpha^{\beta}$ and $\beta^{\alpha}$ are algebraic?

There isn't anything specific about roots of the question. (I was just thinking over).

Of course $e^{\ln2} = 2$, but what can we say $(\ln 2) ^{e}$ ?

If the answer is yes, what can we say about number of such pairs. (Are the set of these pairs finite ?)

After helpful comments let me make this amendment. $\alpha > \beta$.