I have an opaque 6-sided die. Without using mirrors or any other optical aids, what is the greatest number of faces of the die that I am able to see at the same time?
Due to the fact that the die is smaller than the distance between the eyes, if you hold it die very close to your face, it's possible to see 5 faces at the same time (all faces except the one opposite to you):
You didn't specify the material the die can be made of. Dice made of translucent plastic are common, and you can easily see all six sides.
If your goal is to be able to see the most number of sides of a normal opaque die without any sort of aid, whatsoever, besides your own hands and eyes, then i'd say w0lf's answer is the best.
However if your goal is to see the most amount of sides of a normal opaque die without the aid of optical tricks but by utilising other tricks, then i'd suggest simply cutting off the faces of all six sides, thinly, line them up and you will be able to see them all, with an added bonus - the insides of the die! :O