I'm not sure whether I would classify Woodrow Wilson as the most radical President compared to the politics of his time, but he is absolutely a front-runner, at least in the 20th century. At the start of his term he presided over an America that, while definitely finding its feet in the empire department, had not yet been a part of global affairs in the same way as major European powers like Britain, France, and nascent Germany. The First World War, which Wilson campaigned to enter, is considered by many to be the final nail in the coffin of an isolationist USA that was more familiar in the 19th century, and set the stage for further interventionism under Franklin Roosevelt which culminated in, of course, the Second World War.

The departure from isolationism was not, by itself, incredibly radical - most famously Theodore Roosevelt had sought to bring America into both the empire game and global affairs at large, but the practice was also employed by Taft and McKinley at either ends of Roosevelt's presidency. The truly radical elements of Wilson's presidency were changes in the law such as the Espionage Act and the Sedition Act which constrained individual freedom at unprecedented levels under the banner of wartime necessity. Communist leaders were persecuted under his presidency in the First Red Scare at a time when millions of Americans supported left-wing causes.

Wilson's unpopularity peaked towards the end of his term when he attempted, and failed, to bring the US into the League of Nations, a precursor to the UN. After the war, a surge in isolationist sentiment pushed Wilson's internationalist views far into the fringes and undoubtedly lead to a Republican victory in 1920.

My favourite source on Wilson at the moment is: Berg, A. Scott, Wilson (New York: Putnam, 2013) but there are many other texts that support the view that Wilson was a radical president.

Depends on what you mean by "radical" and "the politics of their time." I'll assume you mean departing from the dogma of his party, as well as the wider contemporary political landscape's dogma.

The Roosevelts are both interesting cases for different reasons. With T.R., we witness an extreme political outsider (and something of a chaos entity within his party) rise to the Presidency with the assassination of McKinley, who was ideologically a lot stodgier (what we today like to call 'establishment') than T.R. Roosevelt had been something of a persona non grata within the larger Republican Party as the leader of the Progressive wing. He was sort of your classic smart aleck: young, brash, smart, tough, ambitious, charismatic, unpredictable. Republican leadership rightfully felt threatened by him, thinking him difficult to control.

As President he chose to wield his power with the certain aim of enacting a progressive agenda, marking the dawn of what we now call The Progressive Era. This was something of a political gamble at its conception, though not overwhelmingly so, as it was unpopular with the dominant wing of his party but was backed to some degree by popular mandate on the heels of the Gilded Age. This is why defining the above terms is important, because while the Progressive Era was considered capital-R Radical, it wasn't as though Roosevelt had gone it alone and created a one-man movement. It was not a dominant ideology at the time, though it wasn't lunatic-fringe either.

FDR is a different case. His 1932 campaign was not, as one would assume, swaddled in the rhetoric of the progressivism that he would later champion. If you stripped the name attributions from stump-speech quotes from both Roosevelt and Hoover during the campaign, you'd find it hard to pin down who said what. Roosevelt ran a safe campaign loose on the specifics, and made hardly any promises or explicit embraces of progressive programs before the election; Hoover, for his part, knowing likely that he had little to lose from talking more boldly than his actions had been, made decidedly progressive appeals. It wasn't until the First Inaugural and the First 100 Days that the New Deal would be outlined and enactment would begin, thus beginning Roosevelt's series of more radical programs.

Whether you want to maintain the New Deal was a 'radical' program is somewhat of a matter of debate--Keynesian stimulus would become, of course, arguably the dominant economic antidote domestically and internationally over the two decades that followed, and the concepts of the welfare state and public works of course were no new things. Of course, on the other hand, Keynesian theory was not well-known nor embraced widely until 1936, when Keynes' General Theory was published. At any rate, 'radical' is not a word that you'll often see associated with FDR, mostly because his programs were by-and-large supported by his party leadership and were founded in an emerging dominant economic theory; that is, he wasn't taking a major political risk, nor was he haphazardly breaking from orthodoxy.

This comment is already long enough, so I'll end with a briefer note on Reagan. Reagan indeed was embracing a non-dominant ideology -- one could even safely call it 'fringe' for its degree of aberration from orthodoxy -- and was doing so from the vantage of a definite political outsider. Reagan had the political capital and will to enact this 'radical' economic program with the wind at his back of a stagnating economy that had been overseen by a Democratic Party that had seen its New Deal Coalition finally begin to reach its vanishing point. Carter was the final casualty of a coalition that had fractured and splintered since the 1960s, with Reagan as the direct beneficiary. The Laffer Curve and the broader 'Reaganomics' program were radical in their departure from orthodoxy, both within the party and across the spectrum. Republicans, however, had certainly chosen the right moment to deploy the program, capturing lightning in a bottle with the confluence of the ascension of a charismatic messenger (Reagan), the collapse of the New Deal Coalition (which then, of course, lead to a massive migration of traditional Southern Democrats to the Republican Party), and the natural rebound of the world economy in the early 1980s (stunted once by recession before a strong recovery, which the Reagan White House would spin as the outcome borne of its once-'radical' economic program.)

There are of course others if you want to go back much further, but I just wanted to address the few that you mentioned yourself in the post. One of the cool things you'll learn when you study the presidency and the presidents themselves is that virtually every president put his own stamp on things, and no two presidencies are like, and almost every one will do at least a few little things that make you say "hey, that was kind of ahead of its time."

Edit: I should probably include sources, but most of this is pretty well-known stuff, or easily googled. I could provide links to various readings from my undergrad days, when I specialized in presidential history and the Office of the Presidency, if you're curious about further reading!

Hands down it was Lincoln. He was so radical, in terms of what had been the political consensus prior to his election, that the Southern states seceded rather than accept his Presidency.

Equally as important as ending Slavery (if such a thing can be said...) was the fact that Lincoln firmly and forever established the US government as primarily Federal. Ever since Washington was President, there had been a strong anti-Federal stance (the Federal viewpoint being personified by Alexander Hamilton) taken by Jefferson, Madison, and pretty much every President before Lincoln. Henry Clay was the leading pro-big Federal government politician of the first half of the 19th Century, and the anti-Federalists like Jackson and Calhoun did their best to prevent Clay from achieving the Presidency. Lincoln was the one and only President of the US to utilize Federal troops to forcibly subdue dissenting political viewpoints by gunpoint, ending the various compromises between the geographical sections that had kept the Union together up to 1861.

The America that we have today would not have been possible without a President who was totally committed to imposing his (and his parties') political will upon the entire country (Jackson did threaten to invade South Carolina but that state backed down before things came to blows). Lincoln suspending Habeas Corpus and threatened to jail the Chief Justice of the Supreme Court, effectively assuming dictatorial powers for the extent of the war. After Lincoln, there was no longer any doubt that the primary and chief Governmental power in the US was the Federal Government - not the States - and that the Executive Branch would hold the greatest power.

My money would be on FDR for his radical socio-economic departures and then the gargantuan mobilisation of national resources for victory in WW2, but even he didn't preside over a Civil War. As a President deemed (rightly or otherwise) so "out there" that a large part of the country was prepared to secede and wage war against his Administration for four years, Lincoln must surely be a candidate, a perception reinforced by his determination to see the matter to a resolute conclusion. Like both Roosevelts, he entered office with no intention of carrying through the changes associated with his Presidency: on that scale Reagan is perhaps the frontrunner, though like his predecessors the consequences of his term may far exceed the transformations he had in mind.

I always felt like Jackson was pretty radical, his aggression towards the central banking system was a major chapter in early american politics. But many early presidents had radical or lofty ideas, but they were a part of a very young america, where precedent was not established, lincoln was pretty polarizing obviously, and after him there are the two roosevelt presidents. I would also say jfk and nixon were pretty polarizing in many aspects of their agendas as well. Jimmy carter was another somewhat radical president in terms of his agenda, largely forgotten and erased by reagan and the 1980s conservative revival though.

I think the answers to this question can be more varied than the typical choices, so many different eras with different ideologies and realities, this is an interesting question