Again, Bricklink can help with the answer: When downloading the catalog, you specify "Part and Color Codes". This will give you a list of all part numbers and the color codes they have been produced in. To answer the question, some scripting has to be done to count lines with equal parts, etc.
To make it short, this is the top 10 list:
- Duplo Brick 2x2 (53 colours)
- Slope 30 1x1x2/3, tile 1x2 with Groove, Brick 1x1 (51 colours each)
- Plate 1x2 (50 colours)
- Brick 2x2 (49 colours)
- Brick 1x2 (47 colours)
- Plate 1x1 (44 colours)
- Brick 2x4 (43 colours)
- Plate, round 1x1 straight side (42 colours)
So, it was a close race with a surprising winner from planet Duplon. Looking at the entries I realise that Chrome colors seem to be missing from some parts. They are probably not listed when they don't appear in regular sets (custom parts).
The colour number distribution is very skewed. More than 80% of all 16500+ parts come in only one colour. This is because every printed part gets its own number. There are 2248 minifig torsos, most of them numbered 973pb.....
Anyway, it is fun to browse the catalog.