  The nature of the superconducting transition in highly underdoped thick films
of La$_{2-x}$Sr$_x$CuO$_4$ ($x=0.07$ and 0.08) has been investigated using the
in-plane transport measurements. The contribution of superconducting
fluctuations to the conductivity in zero magnetic field, or paraconductivity,
was determined from the magnetoresistance measured in fields applied
perpendicular to the CuO$_2$ planes. Both the temperature dependence of the
paraconductivity above the transition and the nonlinear current-voltage ($I-V$)
characteristics measured across it, exhibit the main signatures of the
Berezinskii-Kosterlitz-Thouless (BKT) transition. The quantitative comparison
of the superfluid stiffness, extracted from the $I-V$ data, with the
renormalization-group results for the BKT theory, reveals a large value of the
vortex-core energy. This finding is confirmed by the analysis of the
paraconductivity obtained using different methods. The results strongly suggest
that the characteristic energy scale controlling the BKT behavior in this
layered system corresponds to the superfluid stiffness of a few layers.
