A website that would find me a train from Basel (Switzerland) to any city, and then to Berlin for the lowest cost

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Since you are starting from Switzerland, if you book far in advance (or you are lucky), you can get to anywhere in Germany for as little as EUR 39 with an Europa-Spezial-Ticket (it's a non-changeable fare valid only on a specific train and the website also mentions an additional 25% discount on that for passengers with a discount card). On most distances, that would beat anything else, including internal discount fares or regular tickets with a rail card discount.

The Schönes-Wochenende-Ticket (mentioned in the comments) and the Quer-durchs-Land-Ticket or regional discount fares give you more flexibility but they are not valid on long-distance trains and only really cheap when travelling with a group. One big advantage is that they are still available on the day of travel itself. Costs are EUR 44 for up to five people (so obviously much cheaper for a group but slightly more expensive than the best Europa-Spezial fares if you are alone).

The Europa-Spezial fare does not depend directly on distance (tickets to or from German stations close to the border costs EUR 19 or 29 but beyond that the price stays the same) so if you want to save money, you should use it to cover most of the distance and get as close to Berlin as possible. One approach could be to go all the way to some place in Brandenburg or Mecklemburg-Vorpommern and then use a cheap ticket or discount fare to reach Berlin afterwards (a Brandenburg-Berlin ticket works like a Schönes-Wochenende or Quer-durchs-Land ticket but only in Brandenburg and Berlin and can be had for EUR 29 or even EUR 22 if you only travel after 6 PM).

Or maybe use the Europa-Spezial ticket to go to Hamburg, Dresden or Leipzig and then a regular ticket to Berlin (a bit more expensive of course but potentially more interesting if you want to use this opportunity to discover Germany). Regular price from Hamburg to Berlin (second class in high-speed trains) is EUR 78, but there is also an EUR 45 non-changeable discount fare if you book in advance (and possibly a 25% discount on that as well).

Finally, you could also go to a city out of Germany (like you suggested in the comments) to get two Europa-Spezial ticket. In Basel, the Badischer Bahnhof counts as a German station so a trip like Basel Bad-Brussels would be a trip from Germany to Belgium, eligible for this special fare.

To try other combinations, you can use bahn.de. There is also a search widget called the Sparpreis-Finder (“discount fare finder”) to compare available fares easily if you are flexible on the time.

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