The wager came as the result of an Industry Leaders' Challenge begun by the Grand Prix Trust, a charity that helps support anyone who has worked in F1 and who has fallen on hard times.
Wolff nominated Brown to do something he was scared of, and that prompted the idea of a tattoo because the McLaren chief famously hates needles.
Ahead of the United States Grand Prix in Austin, Brown visited a local tattoo parlour and was seen being painted with a tattoo that pays homage to Daniel Ricciardo's recent victory for McLaren at Monza.
Although the final design has not been revealed, glimpses of the tattoo concept suggest it features a map of the Italian GP circuit and the date of the race.
Asked by Motorsport.com about the back story to the tattoo, Brown said: "Sometimes you have to face your greatest fears, which I've done.
"It actually came about through the Grand Prix Trust with Sky, where a team boss gets nominated each year to do something that terrifies them.
"Toto nominated me last year, and needles sufficiently terrify me. So in trying to think through what would be kind of fun and genuine, and properly terrifying, I came up with the idea of getting a tattoo.
"Then obviously, given that Daniel himself is a fan of tattoos, and with him winning recently, I thought maybe we can tie the two together and have him come to visit and hold my hand!"
The nature of the Grand Prix Trust challenge means that Brown can now nominate another high-profile individual from F1 to do something themselves next year.
But Brown was staying tight-lipped on who he had chosen.
"I know who it is, I've done it," he said. "It's to nominate someone significant, a personality in the sport.
"So far it's been three team bosses in a row, but it's not limited to just that. It could be engineers, it could be drivers, it can be other people with significant personalities if you'd like within the sport.
"It's up to the individual to come up with their own plan, and I think it will be rolled out shortly who I've nominated. I'm not allowed to say. It's embargoed."
Brown's tattoo came on the same day that he handed Ricciardo his winning bet for the Australian getting a McLaren podium this year.
Having elected to steer clear on a tattoo on that one, Brown agreed to give Ricciardo a run in a former Dale Earnhardt Sr. NASCAR, which happened on Saturday morning in Austin.