**Abstract** : We study the random sequential adsorption of $k$-mers on the fully-connected lattice with $N=kn$ sites. The probability distribution $S_n(s,t)$ of the number of $k$-mers $s$ covering the lattice at time $t$ is obtained by solving the associated master equation. Taking the scaling limit, we show that the fluctuations of $s$ are Gaussian, with a mean value and a variance both growing as $n$. The probability distribution $T_n(s,t)$ of the time $t$ needed to cover the lattice with $s$ $k$-mers is deduced from its generating function. In the scaling limit, when $n-s=O(n)$, the mean value and the variance of the covering time are both 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 $u=n-s=O(1)$ 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 new scaling regime the master equation governing the evolution of $T_n$ leads, for each value of $k$, to a difference-differential equation for the corresponding extreme-value distribution, indexed by $u$. Explicit results are obtained for monomers (generalized Gumbel distribution) and dimers.