How many forms are there for counting things?

Exactly two, you'll be glad to hear, one for people, viz.:
duine
beirt
triúr
ceathrar
cúigear
seisear
seachtar
ochtar
naonúr
deichniúr;

and one for everything else:
aon
dó
trí
ceathar
cúig
sé
seacht
ocht
naoi
deich

You can see that from 3 up, the two forms are related.