What does it take to make my car fully compliant with all EU countries?

The British AA provides a list of recommended items for selected European countries.

Interestingly, they claim that fire extinguishers are only required for cars registered in Belgium.

Some of requirements not typically mentioned:

• winter tires — under some conditions required in Austria, Sweden, Finland and Germany.
• warning triangles — Spain and Switzerland require not one, but two of them.
• fire extinguisher — Poland not only requires you to have it, but to have it in "readily accessible", which means you cannot have it in the trunk. Typically they are put under driver's seat.
• replacement light bulbs — some countries (France, Poland, Czech Rep., Croatia) require you to have replacement bulbs for any of the lights that is legally required.
• First-aid kit; some countries (at least in Austria). It can be bought in any pharmacy.

Portugal

• You need at least 1 fluo vest.
• Warning triangle
• A spare tire is not required, but if you have one it must be in good shape

I am not sure these are mandatory for cars with a foreigner license plate though.

You will need to buy or rent an electronic payment device to be able to use some highways with electronic tolls only.

The EU put out an app with a list of traffic rules in different countries and, under the “Other Specific Rules”, a list of safety equipment that might be required (including safety vests, warning triangles, fire extinguishers, and winter tires requirements). Unfortunately, it's short on the specifics, does not seem more comprehensive than other similar lists (e.g. from automobile clubs) and comes with a disclaimer.

One thing missing in all the answers and lists so far is the fact that devices warning the driver of speed controls ahead are forbidden in a number of countries (sorry link is in Dutch). The details vary from country to country and I don't know if you should realistically worry about it but I guess that to make your car fully compliant you should leave any personal navigation device with such functionality at home (TomTom has a clever work-around: They warn you of broader “danger zone” instead of telling you exactly where the automated speed control is, which is apparently legal, at least in France).

The environmental sticker system in Germany is also being constantly expanded so that it's now forbidden to ride in many city centers without one. You would need one to make your car fully compliant with German rules (if you wand to go to those cities, obviously).

Finally, many countries require a specific sticker to ride on the motorways but I don't know if that's really relevant to your question. Getting a year pass for all European motorways would be quite expensive. It makes more sense to buy a short-term sticker whenever you actually need to use a country's motorways. You would usually see warning signs and be able to buy one in the border area.