  In this paper we prove a quantitative multilinear limited range extrapolation
theorem which allows us to extrapolate from weighted estimates that include the
cases where some of the exponents are infinite. This extends the recent
extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued
estimates including $\ell^\infty$ spaces and, in particular, we are able to
reprove all the vector-valued bounds for the bilinear Hilbert transform
obtained through the helicoidal method of Benea and Muscalu. Moreover, our
result is quantitative and, in particular, allows us to extend quantitative
estimates obtained from sparse domination in the Banach space setting to the
quasi-Banach space setting.
  Our proof does not rely on any off-diagonal extrapolation results and we
develop a multilinear version of the Rubio de Francia algorithm adapted to the
multisublinear Hardy-Littlewood maximal operator.
  As a corollary, we obtain multilinear extrapolation results for some upper
and lower endpoints estimates in weak-type and BMO spaces.
