The richest man in Illinois does not often give speeches. But on a warm spring day two years ago, Kenneth C. Griffin, the billionaire founder of one of the world’s largest hedge funds, rose before a black-tie dinner of the Economic Club of Chicago to deliver an urgent plea to the city’s elite.

They had stood silently, Griffin told them, as politicians taxed too much, spent too much and drove businesses and jobs from the state. “It is time for us to do something,” he implored.

In the months since, Griffin and a small group of rich supporters — not just from Chicago, but also from New York City and Los Angeles, southern Florida and Texas — have poured tens of millions of dollars into the state, a concentration of political money without precedent in Illinois history.

Their wealth has forcefully shifted the state’s balance of power. Last year, the families helped elect as governor Bruce Rauner, a Griffin friend and former private equity executive from the Chicago suburbs, who estimates his own fortune at more than $500 million. Now they are rallying behind Rauner’s agenda: to cut spending and overhaul the state’s pension system, impose term limits and weaken public employee unions.

“It was clear that they wanted to change the power structure, change the way business was conducted and change the status quo,” said Andy Shaw, an acquaintance of Rauner’s and the president of the Better Government Association, a nonpartisan state watchdog group.

The families remaking Illinois are among a small group around the country who have channeled their extraordinary wealth into political power. Economic winners in an age of rising inequality, operating largely out of public view, they are reshaping government with fortunes so large as to defy the ordinary financial scale of politics.

In the 2016 presidential race, a New York Times analysis found last month, just 158 families had provided nearly half of the early campaign money.

Many of those giving, like Griffin, come from the world of finance. The Florida-based leveraged-buyout pioneer John Childs, the private equity investor Sam Zell and Paul Singer, a prominent New York hedge fund manager, all helped elect Rauner, as did Richard Uihlein, a conservative businessman from the Chicago suburbs.

Most of them lean Republican; some are Democrats. But to a remarkable degree, their philosophies are becoming part of a widely adopted blueprint for public officials around the country: Critical of the power of unions, many are also determined to reduce spending and taxation, and are skeptical of government-led efforts to mitigate the growing gap between the rich and everyone else.

“There was never so much money behind these efforts,” said Iris J. Lav, formerly a senior adviser at the Center on Budget and Policy Priorities, a left-leaning economic think tank in Washington.

To bring about a revolution in the Illinois Capitol, in Springfield, Rauner and his allies have created what amounts to a new campaign economy. Contributing millions to his own campaign, Rauner triggered a state law that removes limits on campaign contributions when a wealthy candidate spends heavily on his or her own race.

The law, intended to limit the influence of the wealthy by providing a level playing field, had the opposite effect: Freed of the restraints, supporters of Rauner poured millions more into his campaign, breaking state records. “He didn’t have to play by the same rules as other candidates,” said Bill Hyers, the chief strategist to Quinn. “He just kept on spending.”

Attacks on Rauner’s wealth fell flat, even as he splashed around money in flamboyant ways: Late in the campaign, he drove up to a credit union on Chicago’s predominantly black South Side, depositing $1 million to support small-business loans.

“It had never happened before,” said Otis Monroe, a community activist in Chicago. “We said, ‘If you want black votes, you should invest in African-American-owned initiatives.’ Rauner was the only one who responded.”

On Election Day, Rauner won every county except Cook County, which encompasses Chicago. That evening, he giddily declared to his supporters: “This is our time. This is a transformational period. We will not accept the status quo. We are going in a new direction — the voters have spoken.”

But voters seemed torn. Along with electing Rauner, they gave Democrats a supermajority in both houses of the legislature.

Rauner was undeterred. Immediately after taking office, he unveiled a strikingly ambitious policy agenda, one with a more ideological tinge than even some Republicans had expected.

Along with expected cuts to spending and property taxes, he proposed tort reform; local “right-to-work zones,” where union membership and dues would be voluntary; and a half-dozen constitutional amendments. He sought to bar public unions from making contributions to state lawmakers — state contractors are already barred — and in February issued an executive order prohibiting public employee unions from collecting mandatory fees from state workers who are not members.

Rauner and his supporters believed such changes were necessary to fix Illinois: Only by disempowering the unions and making the state more hospitable to business, they have argued, can revenue grow fast enough to fix its financial problems.

But despite voters’ deep unhappiness with the direction of the state under Rauner’s predecessor, they quickly soured on their new governor. Just two months into his term, Rauner found that his job approval rate was around 36 percent, according to a poll by the Paul Simon Public Policy Institute. Almost half of Illinois voters favored either tax increases or a combination of increases and spending cuts to fix the budget.

Rauner has since signaled he will discuss new revenues as part of a budget deal, but only if the legislature includes some of his union restrictions or other policy changes as part of the deal.

“I’ve been one who thought he misread his mandate,” said David Yepsen, the director of the Paul Simon Public Policy Institute. “People were ready for a change, but the emphasis on attacking the labor movement, that really poisoned the water here.”

Rauner’s closest supporters hope to elect more Republicans. But some wealthy families, mindful that Democrats are likely to control the legislature for the foreseeable future, have financed an even more ambitious goal: to carve out a new faction of Democrats more willing to reach a compromise with the governor.

That effort has raised more than $14 million, in donations that rival the largest contributions in the presidential campaign. The largest disclosed contribution came from hundreds of miles beyond Illinois: The former Texas energy trader John Arnold and his wife, Laura, gave $5 million.

Arnold, a Democrat, declined to be interviewed for this article. But in an essay published last year, he described himself as a counterweight to traditional interest groups like labor unions and corporations.

“One might ask why Laura and I should be able to influence policy decisions just because we have money,” Arnold wrote. “Were government immune from lobbyists and money, I would agree on the premise of the question. However, government is deeply influenced by special interests.”

His goal, Arnold wrote, was “to counterbalance these entrenched forces, on the right and the left, by providing policy solutions rooted in objectivity and solid analysis.”