Let $Q$ be any II$_1$–factor. It is shown that any standard lattice ${\euscript G}$ can be realized as the standard invariant of a free product of (several) rescalings of $Q$. In particular, if $Q$ has fundamental group equal to the positive reals and if $P$ is the free product of infinitely many copies of $Q$, then $P$ has subfactors giving rise to all possible standard invariants. Similarly, given a II$_1$–subfactor $N\subset M$, it is shown there are subfactors ${\hat N}\subset{\hat M}$ having the same standard invariant as $N\subset M$ but where ${\hat M}$, respectively ${\hat N}$, is the free product of $M$, respectively $N$, with rescalings of $Q$.