Largest semigroup of "nonpermutational" transformations

Let Tn be the set of all transformations of [n]={1,...,n} for each n>0. Tn is a semigroup with product being function composition, i.e. (fg)(x) := g(f(x)) for each f,g ∈ Tn and x ∈ A. A subsemigroup of Tn is a subset of Tn which is closed under composition.
Call a transformation f: A→A nonpermutational if f(B)={f(x): x∈B}=B implies |B| ≤ 1 for any B ⊆ A. (Equivalently, if A is finite, then f is nonpermutational if it has a unique fixed point Fix(f) and no more cycles saving the loop on Fix(f).)
The nonpermutational transformations of [n] do not form a semigroup. The question is, if T is a subsemigroup of Tn consisting of only nonpermutational transformations, how large can T be?

Partial results so far:

• For each n, there exists a semigroup of nonpermutational transformations having size ⌊e(n-1)!⌋. See http://arxiv.org/abs/1203.2873.
• We can show that the size of any such semigroup is at most n!. See http://arxiv.org/abs/1304.5714.

Probably there will be a follow-up on this subject soon -- stay tuned... ;-)