Which character speaks the most languages?

[Inderjit Sanghera]

Few of the Eldar ever learned to speak Valarin...Feanor indeed, before his discontent, is said to have learnt more of this tounge then any others before his time...but what he knew he kept to himself

HoME 11; Quendi and Eldar

This also shows Feanor's genius in which two able loremasters, Rumil and Pengolodoh were in awe (In a way) of him, when writing this, and mastering Valrin was difficult even for the Noldor. I can post some of the 'real' names of the Valar if you want.