Does anyone have a count of the total number of different United States coins regularly minted? This would include all denominations, designs, dates, mint marks, compositions and the like, but would exclude patterns, trials, proofs and the like. Perhaps someone already has this information before I sit down with a Red Book and start counting.

Those old Mint reports have the information. The 1970 report covers from 1793 to 1970 by mint, denomination, etc. There may be some later ones as well, but nothing in this century (they stopped putting in older data a long time ago).