bitcoind seems to respond to a getblock request pretty slowly, so it will take a while to scan the entire blockchain, but so far after scanning backwards through the 1500 most recent blocks, the one with the most transactions has been https://blockchain.info/block-index/388153/00000000000000a8dcc7b7f534b74c0e7e43686407e687217a1d10b40f0bea32 with 1,323 transactions. I'll post another update later when the script has made more progress.
Note: I am only scanning the current best chain. Orphaned blocks will not be included in this analysis.
Thanks for doing this! You should post your findings somewhere and collect donations! (I'll donate something if you do)