The Hoops, who have won the Scottish Premiership and Scottish League Cup this term, added the Scottish Cup on Saturday thanks to a 2-0 win over Motherwell at Hampden Park.
Celtic completed a clean sweep of domestic silverware in 2016-17 and repeated the feat this season, making them the first British club to do so.
"This is the best feeling that you can ever experience," Brown told BBC Sport. "Look at these fans, who turn up every week. Hampden seems to be a special place for us.
"A double treble - I don't think anyone would have had that on the cards when the manager first came in. But it just shows you where hard work can get you. We have made the Champions League [group stages] for the last two years, and we have got to do that again next year.
"We have to go again. It all starts again in four weeks. Some players will be going to play international games but the rest of us will chill and recharge the batteries. We will come back fitter and stronger and I am sure there will be a few additions for next season too."
Celtic's Scottish Cup triumph this season makes them 38-time winners of the trophy.