set theory - Does the beth function have fixed points of arbitrarily large cofinality?

Yes, the κth fixed point will have the same cofinality as κ, since the κ many earlier fixed points are cofinal in κ.

More generally, for any function f from the ordinals to the ordinals, it is not difficult to see that the collection of ordinals α for which f " α subset α, that is, the closure points of f, will form a closed unbounded class C. And for any limit ordinal β, the β-th element of C has the same cofinality as β, since it is the limit of the β many smaller elements of C.

In particular, if we consider the beth function, where α maps to Beth_α, then β is a closure point of this function if and only if Beth_α < β for all α < β, and this implies Beth_β = β. So the club C in this case consists of the Beth fixed points.