  We study the random sequential adsorption of $k$-mers on the fully-connected
lattice with $N=kn$ sites. The probability distribution $T_n(s,t)$ of the time
$t$ needed to cover the lattice with $s$ $k$-mers is obtained using a
generating function approach. In the low coverage scaling limit where
$s,n,t\to\infty$ with $y=s/n^{1/2}={\mathrm O}(1)$ the random variable $t-s$
follows a Poisson distribution with mean $ky^2/2$. In the intermediate coverage
scaling limit, when both $s$ and $n-s$ are ${\mathrm O}(n)$, the mean value and
the variance of the covering time are growing as $n$ and the fluctuations are
Gaussian. When full coverage is approached the scaling functions diverge, which
is the signal of a new scaling behaviour. Indeed, when $u=n-s={\mathrm O}(1)$,
the mean value of the covering time grows as $n^k$ and the variance as
$n^{2k}$, thus $t$ is strongly fluctuating and no longer self-averaging. In
this scaling regime the fluctuations are governed, for each value of $k$, by a
different extreme value distribution, indexed by $u$. Explicit results are
obtained for monomers (generalized Gumbel distribution) and dimers.
