It's easy to find a set of letters of which can make as many meaningful anagrams as there are letters. Such as

one letter: a
two letters: no + on
three letters: apt + pat + tap

etc. Things get more interesting with more letters. So the challenge is..

find a set of n letters which can be arranged to form n+1 meaningful anagrams.

As extra challenge, how about n+2 anagrams?

part
trap
tarp
prat
rapt

Not much of an extension from your 3 letter 'apt' anagrams.

SILENT
ENLIST
TINSEL
LISTEN
INLETS

post, spot, opts, pots, stop is another n+1 for n=4.

aper pare pear rape reap (n+1)

apers pares pears rapes reaps spear spare parse (n+3)

talzamir: don't forget tops for (n+2)

Definitely tops for you. And with this, I suppose it's n+4.🙂

Asper Meaning and Definition
(a.) Rough; rugged; harsh; bitter; stern; fierce.
(n.) A Turkish money of account (formerly a coin), of little value; the 120th part of a piaster.
(n.) The rough breathing; a mark (/) placed over an initial vowel sound or over / to show that it is aspirated, that is, pronounced with h before it; thus "ws, pronounced h/s, "rh`twr, pronounced hra"t/r.