  The tidal deformability of a self-gravitating object leaves an imprint on the
gravitational-wave signal of an inspiral which is paramount to measure the
internal structure of the binary components. We unveil here a surprisingly
unnoticed effect: in the extreme-mass ratio limit the tidal Love number of the
central object (i.e. the quadrupole moment induced by the tidal field of its
companion) affects the gravitational waveform at the leading order in the mass
ratio. This effect acts as a magnifying glass for the tidal deformability of
supermassive objects but was so far neglected, probably because the tidal Love
numbers of a black hole (the most natural candidate for a compact supermassive
object) are identically zero. We argue that extreme-mass ratio inspirals
detectable by the future LISA mission might place constraints on the tidal Love
numbers of the central object which are roughly 8 orders of magnitude more
stringent than current ones on neutron stars, potentially probing all models of
black hole mimickers proposed so far.
