Is there any chance Robert is Jon Snow's father?
I highly doubt that. George R.R. Martin has been laying out the hints regarding Jon's lineage for a while. And one of the key things he asked David Benioff and Dan Weiss when they initially met was who did they think Jon Snow's mother was? Apparently, they answered correctly.
I know that there's sometimes conflict between the books and the television series, but an important thing occurred to me when I was watching the season 1 DVD's recently. During the episode "The King's Road", there are two notable scenes where Ned discusses Jon's mother. First, as Ned parts ways with Jon as they both leave Winterfell, and again when Ned and Robert are talking over lunch.
Clearly, Ned has something to tell Jon, but does not want to tell him that early on in the book. Perhaps he wants to wait until after Jon has taken his vows, and is a sworn member of the Night's Watch? After all, Maester Aemon, one of only three known surviving Targaryen's is a member of the Night's Watch. But as a sworn brother, Maester Aemon has relinquished claims to any titles. Does Ned intend to tell Jon about his true lineage after he is a sworn brother so that he can hold the same status as Maester Aemon?
And then there's the conversation with Robert. King Robert specifically asks Ned about "his bastard's mother" during their conversation along the side of the King's Road. Ned identifies the mother of Jon Snow as "Wylla". But what if Jon's mother is indeed Lyanna Stark, and Ned said as much? That might drive a serious wedge between Robert and Ned, and put Jon in danger. After all, Robert and Lyanna had been set to be married. And it was Rhaegar Targaryen's abduction of Lyanna that caused Robert and Ned to go to war with the Targaryens. Ned is well aware of Robert's hatred for the Targaryens. What would Robert do if he came to learn that Jon was not in fact Ned's bastard son, but the son of Rhaegar Targaryen and the woman he loved, Lyanna Stark? Ned is one of the most honorable characters of the series, but it seems quite clear that he's been lying to everyone about Jon in order to protect him.