Aside from the periods of his presidency when he was in Washington, there simply is not strong evidence to support that. Washington's most regular contacts were his frequent guests at Mount Vernon. Those were typically his next door neighbors. Up until around 1790 when they had a political fallout that included George Mason. But the longest and most continuous was, as I noted previously, Bryan Fairfax. Fairfax and Washington were close friends and confidants for practically 40 years. They corresponded regularly and visited each other from roughly 1760 to 1799, with the last being only a few days before Washington died. At various points well within this period Washington did have frequent contact and correspondence with Hamilton in a largely political capacity. But that friendship cannot be said to surpass the other in closeness, duration, or continuation under even the most strained loyalties.
I would doubt if Washington exchanged more letters with any man than Hamilton and vice versa.
This is doubtful. A quick search of the Washington papers reveals that the two corresponded regularly from the mid 1760's until Washington's death. The earliest is dated 1754 and the latest only a few days before Washington died. This also includes several dozen letters on average for the later years of his life and maxes out the LOC's search engine, which only pulls up to 100 documents. A search for Hamilton reveals a large number of letters between 1795 and 1799 with several dozen a year, but very little before the mid 1780's and absolutely nothing before a few military correspondences in the revolution. At that time Fairfax and Washington had been writing each other regularly for almost 20 years.