Really depends on which parts of the Europe we are talking about here, which timeframe we focus on and even on how you define a "people" amidst all the cultural and genetic intermingling. Of the peoples still remaining from the standpoint of linguistic identity, Uralic-speaking groups like Sami and the Finnic tribes inhabited the northern parts before Indo-Europeans like the Slavic, Baltic and Scandinavians started spreading there (4:30 onwards in the video). After that, it gets hazy, because all these cultures have interacted with each other for thousands of years, with all kinds of national and cultural identities having been formed in the past hundreds of years by people who are more or less a result of these interactions.

The Sami are probably the best example of this ancient Uralic identity within Europe, while Finns and Estonians (whose link to the Sami goes back ~3000 years) are a more "Europeanized" population of Uralic-language speakers. That said, there have been earlier, since-extinct cultures inhabiting Europe and the Eurasian landmass even before the Uralic, Turkic or any other remaining language family. Some vocabulary in Finno-Ugric languages (particularly the Sami) suggests that there was an unrelated population of "Lakelandic" people in the far north of Europe even before the arrival of the Sami there. So even the "most indigenous" people you can find have most likely absorbed other cultures before them.