Why did the other Continental European powers keep trying to fight Napoleon before 1813 despite being beaten by him so often?

• There was one king/tsar that tried to ally with Napoleon, Pavel I of Russia. He was killed.
• His son Alexander I once tried to make friends with Napoleon. He was threatened and broke the agreement.
• Swedish King Bernadotte was one of Napoleon's Marshals. But he started to fight against him, too.

All Napoleonic wars were about struggle of French and English elites for the Europe domination. The whole Europe was already under much stronger English economical influence. So, the elites of all continental countries (except France) did not want to boycott England. And not boycotting England meant war with Napoleon. Kings could have any views, but they could not ignore the interests of all of the elite in their countries. Pavel tried...

So, several years repeated the same cycle: war against France, defeat, peace... Napoleon won military, but again and again lost economically and consequentally, in politics. His enemies among the elite of the questioned country overpowered his allies. And a new war was starting.

There were two possible ways out of that cycle: Economic victory of the French elite (never happened) or the military defeat of France (which finally happened).

One more argument for my point of view: Notice, that England did not participate in that peace/war circling - it was the constant enemy on France these days.

The Napoleonic Wars are perhaps best thought of as a single continuous conflict between Britain and France, extending for twelve and a half years from the breakdown of the Treaty of Amiens through Napoleon's defeat at Waterloo.

Now, Britain at the time didn't have much of a standing army to fight with. Rather, British strategy was to use their superior navy to control the seas and cut off their opponent's access to overseas colonies, while relying on Continental allies to engage in land combat. Diplomatically, their goal was "balance of power": arranging alliances to keep any single Continental country or group of countries from becoming too powerful.

You can see how this plays out in the Napoleonic Wars: the "War of the Whichevereth Coalition" is simply Britain's latest group of allies in an ongoing conflict.

Napoleon represented an existential threat to the other rulers of Europe.

First, although he was technically an Emperor of France, he was not descended from royal blood like all the other emperors, kings, and princes. He was an "upstart" who had crowned himself, both literally and figuratively.

Putting aside "social" issues, he was highly disruptive to the politics of Europe. By putting his brother on the throne of Spain, he renewed fears of a Franco-Spanish behemoth. By creating the "Confederation of the Rhine" (a group of client states encompassing mostly the modern "West Germany"), he created competition for the Austrians for "Holy Roman Emperor." He reduced Prussia to half of its former size by creating the Duchy of Warsaw (Poland) and some west German states out of its territory. He also took away Austrian lands in Italy and what later became Yugoslavia.

The Austrians and others never wanted "peace" with Napoleon, only truces that would let the reorganize and rise again. They did so when Napoleon suffer reverses, e.g. against the British in the Peninsular War in Spain, and, of course, later against Russia.

There are some assumptions you've made, which are all too easy to do when judging historical events from a modern lens:

• Hindsight is 20/20. We know that Napoleon would beat them again and again, but they probably thought "maybe this time!" which they eventually did - the Russian winter, anyway.
• Be careful not to cherry-pick. Napoleon's expansion was checked by the British, twice: at Trafalgar, and in Iberia.
• It's an oversimplification to say they kept fighting Napoleon; often you'll find that there were different people involved and under different circumstances. Even genuine repeat wars occur when the last one did not resolve all differences.

Of the Napoleonic wars, Napoleon won three: the Third, Fourth and Fifth coalitions. Between the Third and Fourth, Prussia entered the coalition. They originally feared Napoleon especially after the latter's stunning victory at Austerlitz, but ongoing disputes finally lead them to breaking point. Between the Fourth and Fifth, France subdued Russia but were spread thin and tied down in Iberia, and Austria sought to take advantage of this and subject France to a two-front war. By this stage the French army was qualitatively weaker, and Napoleon suffered his first personal defeat in a decade at Aspern-Essling.