The Pole was officially confirmed as a Bayern player in July after signing a pre-contract agreement with the Bavarians in January, ending his prolific four-year spell with BVB.
The 25-year-old netted 103 goals in his time at Signal Iduna Park and, ahead of the DFL-Supercup in which he will face his former club Dortmund on his competitive debut, he stressed his belief he can make further strides under Pep Guardiola.
"I know I can get even better. It doesn't matter where you've played and what you've won, you always have to give it everything you've got," Lewandowski told reporters.
"I am not yet at the limit. I hope I can get better every day."
Bayern's stars from Germany's successful World Cup campaign will return to the squad for their last friendly clash with an MLS All-Star team at Providence Park in Portland on Wednesday, just a week before their clash with Dortmund on August 13.
The DFL-Supercup is to be held at Signal Iduna Park and, while Lewandowski is relishing the occasion, the forward expects his first competitive match as a Bayern player to be a challenging one.
"It's obviously going to be special for me," he added. "I had four fantastic years at Dortmund, but it'll feel very different as a member of the visiting team.
"It will be a difficult evening, simply because we'll only have trained all together for a week. But we're a good team and we'll obviously play to win."