  Extreme mass ratio inspirals (EMRIs), i.e. binary systems comprised by a
compact stellar-mass object orbiting a massive black hole, are expected to be
among the primary gravitational wave (GW) sources for the forthcoming LISA
mission. The astrophysical processes leading to the formation of such systems
still remain poorly understood, resulting into large uncertainties in the
predicted cosmic rate of these sources, spanning at least three orders of
magnitude. As LISA can individually resolve mostly EMRIs up to $z\gtrsim1$, the
ensemble of signals below its detection threshold will add up incoherently
forming an unresolved confusion noise, which can be formally described as a
stochastic background. We perform an extensive study of this background by
considering a collection of astrophysically motivated EMRI formation scenarios,
spanning current uncertainties. We find that, for most astrophysical models,
this signal is easily detectable by LISA, with signal to noise ratios of
several hundreds. In fiducial EMRI models -- predicting hundreds of EMRI
detections during mission operations -- the background level is comparable to
the LISA noise, affecting the performance of the instrument around 3 mHz. In
extreme cases, this background can even "erase" the whole LISA sensitivity
bucket in the 2-10 mHz frequency range. This points to the need of a better
understanding of EMRIs' astrophysics for a full assessment of the LISA mission
potential.
