What's the expressive/compressive power of strongly normalizing subset of untyped lambda calculus? Let $\Lambda$ be a set of strongly normalizing lambda terms.
Let $\mathtt{NF} : \Lambda \rightarrow \Lambda$ be evaluation to the normal form.
Let $ \lvert x\rvert : \Lambda \rightarrow \mathbb{N}$ be 
size of a term in number of symbols.
The following function seems to be well defined:

$$f_\Lambda:\mathbb{N} \rightarrow \mathbb{N}$$
$$ f_\Lambda(n) = \max_{t \in \Lambda, \lvert t \rvert \leq n} \lvert\mathtt{NF}(t)\rvert $$ 

What is the rate of growth of $f_\Lambda$?

Can we define some lower and upper bound in fast growing hierarchy?
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.