Q:

"could possibly do something" vs "could do something"

Tell me please what the difference is between the following sentences.

He could possibly win the game.
He could win the game.

What is the role of possibly in the sebtence? What does that adds to the meanning?

A:

In this context, "possibly" actually lends doubt to the statement, and will often be seen with emphasis of some kind, when written.
So, your first example means that there is lets say 0 to 10% chance, while the second means anything from 1% to 100%
The context and the tone of voice used will make it clearer exactly how much difference there is.

