The closed classes of $k$-valued logic $P_k$, $k\geq 3$, which are maximal among all closed classes without finite bases are constructed. Such classes have no finite bases, but all their proper closed super-classes have finite bases. Such classes are called here maximal. It is shown that for any $k\geq 3$ maximal classes exist in $P_k$, and the set of these classes is at most countable. For $k=3$ a maximal class of depth 5 in the lattice $\mathfrak C_{k}$ of all closed classes of $k$-valued logic is found, and for $k>3$ similar classes of depth 3 are described.