Still, Lee said he is working with Democratic senators on bills dealing with the privacy issues raised by recent revelations about the National Security Agency surveillance program.
Although he has also been called an extremist, Lee said that's not fair. He said it's extreme that the government "is increasingly revealing that it has spied on us. It has lied to us. It has targeted us as Americans."
Lee is attracting national attention for his stands. On Sunday, he is scheduled to be a guest on FOX News Sunday to talk with host Chris Wallace about his proposal to stop funding the health care law.
"It's pretty clear he's interested in being a nationally visible leader of the conservative movement," said Matthew Wilson, a political science professor at Southern Methodist University in Dallas.
Wilson said while Lee has yet to get as much attention as fellow GOP conservative senators Rand Paul of Kentucky, Ted Cruz of Texas and Marco Rubio of Florida, that could change if he makes headway against Obamacare.
Paul, Cruz and Rubio are all seen as contenders for the Republican presidential nomination in 2016. Lee said he's happy in the Senate and has no other political ambitions.
Wilson said, however, it's going to be tough for Lee to be able to claim a victory in his fight against the health care law.
"In the end, from a practical, political standpoint, it probably goes nowhere," Wilson said. "It's a long shot bid to do something legislative at this point to undermine Obamacare. But a lot of times in politics, the symbol matters more."
University of Iowa political science professor Tim Hagle, an active Republican, said the GOP could end up energized by Lee's efforts no matter what ends up happening.
He said Lee, who replaced veteran Sen. Bob Bennett in 2010, could gain influence in the Senate as part of a younger generation who see their role as stopping legislation they oppose, just as Lee is doing.
"That can make him a powerful player in the Senate," Hagle said.
Lee said he's not after a more prominent profile.
"I just care about this issue," he said. "The attention isn't about me. The attention is about the cause. It'd be a lot easier to just not express opinions....But I don't think that's what I was sent there to do."
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