The highest scores in professional golf history, including on the PGA Tour, have typically happened not because the player isn't skilled. Rather, the highest rounds in pro golf history have occurred because of injury or in protest.
Mike Reasor has the highest-recalled score in PGA Tour history, dating back to his performance in the 1974 Tallahassee Open. Reasor made the 36-hole cut at 144. After the second round, Reasor went horseback riding. The horse got spooked and ran Reasor into a tree, leaving him with a torn rib cartilage, damaged knee ligaments and a separated left shoulder. Since the Tour wasn't all-exempt back then, Reasor needed to finish the tournament to get an exemption into the Byron Nelson Classic the next week. So, he played golf swinging just a 5-iron one-armed to get through the weekend. He shot 51-over 123 in the third round and 114 in the final round.
Reasor got into the Nelson, but he withdrew anyhow.
On the European Tour, the record seems to belong to former Ryder Cup captain Mark James. In the 1978 Italian Open, scoring conditions were terrible, but James shot 111 in the second round for a different reason. In an effort to protest the European Tour fining him for withdrawing from a tournament, James shot 111 one-armed to prove his point.
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On PGA Tour Latinoamerica, Maurice Allen shot 115 in the 2013 Dominican Republican Open.
None of the majors professional tours keeps a record for the highest round in their history, and, if we wanted to really look for it, we could find the highest round in pro golf history back in the 19th century, when the Open Championship was played with hickories and featheries.