What was the highest ever win for a decision in a nationwide *democratic* voting?

I would submit the Swedish general election of 1968, where the Social democratic party achieved 50,1 % of the (raw) votes, with a voter turnout of almost 90 % (they had achieved even more in 1940, but then war conditions made for some small infractions against the communist party and a lower turnout than usual).

As for referendums, I would cite the Norway referendum for approving the dissolution of the union with Sweden in 1905, but there were less than the required 500 000 voters, and women had no suffrage. Still, it achieved an impressive 99,95 % approval.

Your goal seems to be to establish good bounds for what a reasonable margin for a "big victory" looks like in a national election for a single leader in a functioning Democracy. Not a lot of Democratic countries actually have direct elections for Head of State. The two most prominent I know of are the USA and France.

I know you specified direct election, not electors, so we will ignore the messy issue of the USA Electoral College. Even though that exists as a mechanic, USA presidential elections (after 1866) do have their overall vote totals calculated and publicly available. So we can still look at those, and ignore the details of what the Electoral College decided to do with the votes that were cast. There is still the priviso that we have to start looking after the 14th Amendment, since before then some states simply appointed electors rather than holding an election.

The largest margin of victory since 1866 in a presidential election was a 26.1% margin of victory for Warren Harding in the 1920 election. His actual raw percentage of the vote was 60.3%, which was bested slightly a couple of times. The record holder there is Lyndon Johnson in 1964 with 61.1% of the vote (and a 22.6% margin over Barry Goldwater).

Comments from user @andejons mentioned France, which is one of the few other western countries I'm aware of that has a nationwide vote for a Head of State. Since the foundation of the current French Republic there have been two "blowout" elections. The first one (which we probably ought not to count. The next opponent wikipedians couldn't even dig up a photo for), and the election of 2002, where the opponent was largely viewed as a neo-Nazi. The latter was won by a whopping 64.4% margin (with 82.2% of the vote). That probably takes the cake for the largest victory in an indisputably Democratic direct head-of-state election ever.

Aside from those two exceptional circumstances, the largest ever margin of victory in a French Presidential election (for the current Republic) was in 1969, where 16.4%, with the winner Pompidou getting 58.2% of the vote. Both lower than the USA records, but we have less French elections to use as data points.

Based on this, I think its fair to say that if we are looking at a first-past-the-post national election for a single leader, in a legit Democracy the largest margin of victory you should see should be somewhere in the 20's, and largest total vote % in the 60's. If you see something significantly higher than that, something fishy is going on. Most likely the voters did not think voting for the next best candidate was really an option.

The 2013 Falkland Islands sovereignty referendum yielded a 99.8% for British sovereignty. Runner ups are 1967 Gibraltar sovereignty referendum with 99.64% and the 2020 Gibraltar sovereignty referendum with 98.97%.

That level of support is uncommon because referendums are often deemed unnecessary when its outcome is clear in advance - that is, a parliament would yield the same decision and it would be unlikely that it were contested. Referendums for questions with nearly unanimous support are only held where there is a need to show that unanimous support.

Switzerland practices direct democracy and has many votings.

There is a list of Swiss federal referendums at wikipedia with the results. The results are listed in the German version. A short translation help of the columns:

• Beteiligung = voter participation
• Anteil Ja-Stimmen: Yes-votes.

Some results with high yes vote (>80%):

• 11.03.2012: 87% yes with 43% voter participation Regelung der Geldspiele zugunsten gemeinnütziger Zwecke (regulation of gambling for charitable uses).
• 18.05.2014: 88% yes (54% participation), about primary health care
• 21.05.2006 85% yes (27% participation)
• 2.03.2000 86% (41%): justice reform
• 10.03.1996 91% yes (30% participation) Affiliation of Vellerat to Jura

There is one problem on this numbers and your question: You can only decide yes/no (or invalid?). So all votes are normally shared between two possibilities.

There's an assumption in your question, that free and democratic voting can't result in very high percentage wins. In addition to whether it's free and fair, it also depends on the voting system and the choices being voted for.

There's an important observation in economics known as Hotelling's law that's applicable to political elections. The premise is that:

Suppose that there are two competing shops located along the length of a street running north and south. Each shop owner wants to locate his shop such that he maximises his own market share by drawing the largest number of customers. In this example, the shop itself is the 'product' considered and both products are equal in quality and price. There is no difference in product to the customers. Customers are spread equally along the street. Therefore, considering the prices are exactly the same, each customer will always choose the nearest shop because there is no difference in product.

The law predicts that, when the number of choices is limited to two, those two choices will tend to become similar and towards the middle in order to maximise their voters. This is the reason why two-party voting systems produce two parties that are very much alike, and where the winners tend to win just over 50% of the votes (discounting anomalies like protest votes). You may have heard such observations being applied to the American voting system.

However, when the number of choices isn't limited to two, the "winner" can get even less of the total votes. For example, in the 2013 German federal election, the winning party won 42% of the votes, despite this result being the best in decades for the party.

Another counterexample is referenda, where the choices are fixed and mostly immutable. As the other answers noted, there are many referenda where the winning choice won overwhelmingly.

The French presidential election of 1848 will likely catch your interest. This direct popular vote saw Louis-Napoléon Bonaparte win with 74.44% of the votes. (No idea what the turnout was.) He was the first (and only) president of the 2nd French Republic. He was blocked by the Constitution and by the Parliament from running for a second term, so did a coup in 1851 and took the throne as Napoleon III in 1852. It had a durable impact on French politics: the fear of popular politicians gaining dictatorial control over the country through plebiscite votes led France to avoid direct elections until De Gaulle a century later.

The French Presidential elections of 2002 were a bit peculiar but might also be a good answer for your question. The second round of the two round direct presidential election saw the winning candidate get 82.21% of votes with a 79.71% participation rate.

The latter pitted (right wing) Jacques Chirac, then President, against (far right wing) Jean Marie Le Pen. Chirac was then quite unpopular, but still seen as the lesser of two evils by the country - left included, and its electors basically showed up en mass to form a barrage against Le Pen.

Depending on the commentator, the final score was either commended as a clear message in favor of democracy and tolerance, or mocked as approaching banana republic levels. Needless to say, Chirac had no legitimacy whatsoever after the vote, and the Elysée Palace received a number of unused "Le Pen" bulletins sent by enraged left wing voters to remind Chirac that they voted against rather than in favor.

In the follow-up legislatives elections, Chirac's party went on to successfully campaign against a new period of cohabitation, against a left in shambles (its leader, Jospin, quit politics on the spot) that was campaigning for the opposite. It too was very much a for/against vote.