# 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?