The largest closed set has $2^{10} - 1$ numbers. To get this number, start with ten primes $p_i$, $i = 1, 2, \dots, 10$. The closed set consists of all products $\prod_{i \in X} p_i$ over all subsets $X$ of the set $\{1, 2, \dots, 10\}$, hence there are $2^{10} - 1$ elements in the closed set.

One cannot obtain more than $2^{10} - 1$ numbers. Indeed, each move gives the l.c.m. of two or more numbers from the initial set, so each added number corresponds to a nonempty set of the set $\{1, 2, \dots, 10\}$. The conclusion follows.