  We study a function $\mathcal{L}_{\Gamma}$ which quantifies the LEF (local
embeddability into finite groups) property for a finitely generated group
$\Gamma$. We compute this "LEF growth" function in some examples, including
certain wreath products. We compare LEF growth with the analogous quantitative
version of residual finiteness, and exhibit a family of finitely generated
residually finite groups which nevertheless admit many more local embeddings
into finite groups than they do finite quotients. Along the way, we give a new
proof that B.H. Neumann's continuous family of $2$-generated groups contains no
finitely presented group, a result originally due to Baumslag and Miller.
Finally, we show that there exists a "universal" function which is an upper
bound on the LEF growth of any group on a given number of generators, and that
(for non-cyclic groups) any such function is non-computable.
