Euclidean construction - is it possible?

To anyone who knows a bit more about Euclidean construction than I do (i.e. construction using compass and straightedge):
Is any of the following expressions geometrically constructible? Given a side of length 1 is it possible to construct a side of any of the following lengths:

sqrt(1) (this is trivial, I know it's possible :P)
sqrt(1 sqrt(2))
sqrt(1 sqrt(2 sqrt(3)))
sqrt(1 sqrt(2 sqrt(3 sqrt(4))))

and the same for any n? I'm particularly interested in the case for 4, because I 'kind of' saw it get constructed in a dream lately, but the dream also mentioned that it's possible to construct sidelengths like that for any n.
Is it a coincidence? I kind of know basic rules for what's constructible and what isn't (cube root of 2 isn't, pi isn't, square root of 2 is, etc.) but such expressions like these are still black magic to me.