What's the expressive/compressive power of strongly normalizing subset of untyped lambda calculus?
The maximal blowup in termsize of simply typed lambda calculus is non-elementary (2 ↑↑ O(n) see e.g. [1] page 72). This is not exactly what you are asking for, but since typed lambda calculus is a strongly normalizing subset of untyped lambda calculus, it should give you some lower bounds.

I am not sure if the general question can be answered. Untyped lambda calculus is Turing-complete, the strongly normalizing fragment is then equivalent to the set of terminating programs. Since membership in this set is already undecidable, I would not expect that an upper bound on the size of normal forms can be shown.

[1] Morten Heine Sørensen and Pawel Urzyczyn. Lectures on the Curry-Howard Isomorphism, volume 149 of Studies in Logic and the Foundations of Mathematics. Elsevier Science Inc., New York, NY, USA, 2006.