The three major language branches in Europe are:
- Germanic (for example English, German, Dutch and Scandinavian languages)
- Romance (for example French, Spanish and Italian)
- Slavic (for example Russian, Polish and Serbian-Croatian)
So our “most unique language” would have to be one outside of these branches. Among the bigger languages, it could be Finnish (or its sister languages of Estonian and Hungarian) or Greek.
Among smaller languages Basque is worth noting, because it’s a “language isolate”. It’s not classified into any of larger language families.
Despite belonging to the Romance branch, Rhaeto-Romansh (spoken in South-Eastern Switzerland) deserves a mention, because it’s a rather unique and small language, but retains a (semi-)official status in Switzerland. The reason I find it interesting is that you can address the federal parliament in this language (and get an answer in its standardized version).
While we’re at it, we could throw Faroese and Icelandic in the mix. The reason Icelandic is worth mentioning is that it’s probably the living language closest to Old Norse (the language of the Vikings).