Let D be an integral domain and SF(D) be the set of star operations of finite type on D. We show that if ${mid}SF(D){mid}$ < ${infty}$, then every maximal ideal of D is a $t$-ideal. We give an example of integrally closed quasi-local domains D in which the maximal ideal is divisorial (so a $t$-ideal) but ${mid}SF(D){mid}={infty}$. We also study the integrally closed domains D with ${mid}SF(D){mid}{leq}2$.