The Scottish midfielder has led the Hoops to the past two titles and with a third on the horizon, he wants to go after even more silverware as the green and whites maintain their dominance on the rest of the competition.
Celtic won the 2006, 2007 and 2008 premierships before Old Firm rivals Rangers pinched a fourth from them on the final day in 2009 and Brown is looking to secure a third before adding to that next season and beyond.
"We can see the finishing line. Three-in-a-Row has a nice ring to it. Now we will look to kick on. I don't see why not," said Brown in the wake of Friday's 3-0 win at Kilmarnock.
"The lads we have in that dressing room are outstanding and if we can try to keep the squad together as long as possible we will see where we can go.
"We've won three before but we just didn't quite get to four. It was hard, losing to our biggest rivals.
"It was a killer when Rangers managed to get over the line.
"But with the way it is looking now we need to get over the line with the third one then try to build to four and five. We won't try to look too far ahead though."
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