  Unconventional superconductivity typically occurs in materials in which a
small change of a parameter such as bandwidth or doping leads to
antiferromagnetic or Mott insulating phases. As such competing phases are
approached, the properties of the superconductor often become increasingly
exotic. For example, in organic superconductors and underdoped
high-$T_\mathrm{c}$ cuprate superconductors a fluctuating superconducting state
persists to temperatures significantly above $T_\mathrm{c}$. By studying alloys
of quasi-two-dimensional organic molecular metals in the
$\kappa$-(BEDT-TTF)$_2$X family, we reveal how the Nernst effect, a sensitive
probe of superconducting phase fluctuations, evolves in the regime of extreme
Mott criticality. We find strong evidence that, as the phase diagram is
traversed through superconductivity towards the Mott state, the temperature
scale for superconducting fluctuations increases dramatically, eventually
approaching the temperature at which quasiparticles become identifiable at all.
