Brendan Rodgers led Celtic to an unbeaten domestic treble in his first campaign in charge of the Hoops. The Scottish champions, who face Hearts on Saturday, have not tasted domestic defeat since May 2016.
Brown told 'Sky Sports News', "we want to try and push that as far as we can, every game is going to be hard, especially the opening game of the season. Our fans will be there and will be behind us all".
"We've got to maintain our levels that we played at last season. It was an exciting start last season and we kicked on. Nobody thought we could go all the way [unbeaten], but we managed to do that."
"But that was last season, it's all been forgotten about now and we've got to start fresh again. The first game of the season is against a good old pal, Christophe Berra. I'll be looking forward to him kicking me and it should be a good contest."
Brown is also hoping the Hoops keep hold of star striker Moussa Dembélé. The Celtic captain said, "we want him to be at the club for as long as possible. He's got strength and different sort of qualities. When he played in Champions League matches, he scored and created chances. His hold up play was fantastic."
"Yet again, we've got Leigh Griffiths there and he scored 40 goals the season before. He didn't have as many games last season, but still scored a fair amount of goals."