  We introduce a new iterative amalgamated free product construction of II$_1$
factors, and use it to construct a separable II$_1$ factor which does not have
property Gamma and is not elementarily equivalent to the free group factor
$\text{L}(\mathbb F_n)$, for any $2\leq n\leq \infty$. This provides the first
explicit example of two non-elementarily equivalent II$_1$ factors without
property Gamma. Moreover, our construction also provides the first explicit
example of a II$_1$ factor without property Gamma that is also not elementarily
equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of
techniques from Voiculescu's free entropy theory and Popa's
deformation/rigidity theory.
