It is really not easy to answer the question which is the largest freshwater fish, not only because you have to discriminate between pure freshwater fish, migratory fish and fish which come on occasion in freshwater.
A very big (and I mean really big) problem is that most of dates of maximum sizes you can find for the biggest specimens of a species are extremely unprobably and highly likely to be big-fish-stories. To take for example the arapaima, the only reason why so many people believe in this 15feet is because there was one single report from the first half of the 19th century that this fish should grow to this size (ever realized that a 15 feet long fish with only 200kg would look like an eel...). I wrote a longer post about this topic:
http://www.monsterfishkeepers.com/forums/showthread.php?t=168521
The same thing is with the wels catfish and many other species too. You will find big-fish-stories everywhere, but if you look how big the actual confirmed records were, you see in general a huge difference.
There are big fish stories for many sturgeons too, as well as for the giant freshwater stingrays and many other ones.
Another problem is how to count, length or weight.
If you only count fish which live in freshwater, there are some very big ones (I don´t write them as a list from the biggest to the smallest):
Wels catfish Silurus glanis: Record size 2,78m and 144kg
Piraiba catfish
Brachyplatystoma filamentosum: 2,37 and 142kg
Arapaima : 2,63m and 185kg
Alligator gar: (at least) 2,89m and 165kg
Giant Mekond Catfish: 2,70 and 293kg
Giant freshwater stingray: at leat 3m and well over 100kg
Chinese paddlefish: The often cited report of 7m is only one single report and much larger than anything else, so I would not take it serious. But lengths of more than 3m are confirmed as well a weights of more than 100kg, like the specimen caught some years ago which was 3,30 and weighed 117kg, but actual record sizes would be most probably higher.
There are also some very big other sturgeons which only live in freshwater which are in the 2m range, as well as some other ones.
So if you count only weight, the heaviest freshwater fish with a confirmed weight is most probably the Giant Mekong Catfish, but if you count length there are several species which can reach more length, probably with the chinese paddelfish at the top. This species seems also to be those with the highest average length.
If you count migratory fish you can also count a lot of sturgeons. There are also many figures which are highly dubious like the often cited 8-9m and 1500kg for Beluga sturgeons. Actually the biggest ones which were confirmed were actually about 1500kg in weight (which is really very much over average and as big as a big great white shark), but at this weight they were "only" about as long as 5,5m. A hypothetic 9m beluga would weigh about 6500kg, about as much as a minke whale.
But if you count also species which come only occasional in freshwater, there is possibly even a bigger species.
Sawfish also sometimes swim up in the rivers for example in Australia, and there are many good records of specimens well over 5,5m.
