Where should I travel in Europe if I want to climb a high mountain without equipment?

I did some research and it seems that Inneres and Üsseres Barrhorn in Switzerland are the highest mountains in Europe that can be reached by hikers. This is also mentioned on the English Wikipedia page. The German Wikipedia page has more information about that. On hikr.org, a very popular hiking platform in Switzerland, it is also mentioned that the Barrhorn is the highest moutain which you can reach by walking. Unfortunately the information is only available in German.

In Europe you have several options, often encompassed by the Alps.

While you can use a tour, often the tour suggestions are merely good idea for you to use. But they do at least let you know if it's able to be hiked without any climbing gear.

You can hike in the Italian Dolomites, the Carpathian Mountains, Bulgaria, Transylvania, the Fjords of Norway and more. Some descriptions of these hikes are on Mountain Hiking Holidays.

More on getting to the Tatra mountains in Poland and Slovakia.

Here are the 5 best day hikes in the Swiss Alps.

EDIT:

And if you just want a single mountain, consider the top 10 tallest mountains in Europe.

The German site linked to in Roflcoptr's answer mentions an even higher mountain in France, the Aiguille de la Grande Sassière (3751 m) and claims this is definitely the highest "hiking mountain" in Western Europe, but cautions that the ascent includes some "light climbing" passages, and depending on the weather the final ascent might be ice-covered.

I did something similar last year. From Zermatt a hiker can easily reach the Oberere Rothorn and the Mettelhorn, both 3,400+. The Mettelhorn is probably my best hike ever. 1800 metres ascent, a harmless glacier crossing (had no axe or crampons), and a wonderful view!

My personal record is the Stockhorn from Gornergrat, at 3,532 on the newest map, but I thought it was 3,525 when I did it. The route is annoying in places but it's quite doable, and puts you surrounded by glaciers on three sides.

I have my eye on the Barrhorn for next season.

If you want to climb the highest mountain in the UK, that would be Ben Nevis with ~1344m. There is a city nearby called Fort William and the surrounding area is stunning! Especially going from Fort Williams to the west coast by train is a great ride and there are many lochs and hills to explore.

Alternatively in the Isle of Skye in northern Scotland there lies a mountain range called the Cuillins, it is said that the hardest hikes in the UK go over the Black Cuillins, hard due to the strange shape of the mountains. The landscape on the island is amazing as well. Very thick and old tolkien-esk forests in the south and the rocky Cuillins in the north.

Not quite extreme alpine climbing, but if you want to avoid the people and enjoy a beautiful landscape than this might be the thing for you.

Sweden's highest mountain Kebnekaise is accessible by foot though it is not that high (only some 2106 metres). My co-blogger Hardy did it and wrote about it.

Depending on what you consider the boundary of Europe, Mount Elbrus can be considered the highest point in Europe. At 5642 m, it's considerably taller than Mont Blanc (4810 m). The standard route is pretty nontechnical, and requires only an ice ax and crampons.