But the governmental dysfunction has become a sideshow to an even bigger controversy over Mr. LePage’s actions regarding a charter school for at-risk youths. The school had hired Mark Eves, the Democratic speaker of the House and a LePage foe, as its next president, starting Wednesday. Mr. LePage said Mr. Eves was unfit to lead the school, and threatened to withhold more than $500,000 in annual state money unless the hiring was rescinded; the school, a nonprofit fearing the loss could threaten private matching funds and lead to its closing, did so.

Mr. Eves accused the governor of blackmailing the school and threatened to sue him.

The governor’s actions have infuriated many who say he overstepped his executive authority; a group of Democrats and independents in the Legislature is researching how and whether to impeach him. Democratic leaders are taking a cautious approach, but have said nothing is off the table. An anti-LePage rally is being planned here for noon on Tuesday.

“Things have gone way off the rails, and there’s a real question about whether he’s fit to govern,” said Phil Bartlett, chairman of the state Democratic Party. “I support legislators getting to the bottom of this and making sure there’s accountability.”

Lawmakers say they are exhausted and frustrated.

“When you’re sitting there poking the yes button every 15 seconds to override 128 vetoes, it’s hard to feel like you’re contributing a lot to moving the state forward,” said State Senator Roger J. Katz, a Republican who has been an occasional LePage ally but more often a critic. “The atmosphere here is one of depression,” he said.

For one thing, Maine’s problems, like its lagging economy, are not being addressed.

Mr. LePage declined an interview request. But Rick Bennett, chairman of the state Republican Party, defended the governor as a change agent, someone who provided “a disruptive energy,” in a good way, and who had “shaken Augusta to its roots.”