The minimum number of operations is $$$T=\lceil log_2(K) \rceil$$$.

To print a solution, consider a $$$100 \times 100$$$ grid that have the numbers from $$$0$$$ to $$$K-2$$$ written on the first $$$K-1$$$ cells of the rows with odd index (except in the last column) and the rest of cells have written $$$2^T-1$$$ on them. For example, for $$$K=102$$$:

The idea is to use the machine during the $$$i$$$-th operation on the cells that have the $$$i$$$-th bit set. Every cell is adjacent to at least one cell with the number $$$2^T-1$$$, so the cells used during each operation will form a connected component, and we will have $$$K$$$ types at the end.