Game of Thrones' most widely believed theory — that Jon Snow is actually the son of Lyanna Stark and Rhaegar Targaryen — finally came to fruition in the sixth season's finale. The list of characters that know this secret is likely very short: Bran Stark of course, since it was through his vision that the parentage was confirmed, Howland Reed, who was outside the Tower of Joy and hasn't been seen on the series and Ned Stark, who is dead. Therefore, it's likely not a fact greater Westeros will be made aware of anytime soon.
However, another longtime character on the series might've known about this secret the whole time — someone who Jon confided in before his eventual death. As redditor sakhnini1 noted, Maester Aemon of the Night's Watch, one of few remnants of House Targaryen, could have been hinting at the events that took place at the Tower of Joy while he was counseling Jon.
After Ned is imprisoned in King's Landing back in season one, Jon is understandably frustrated at what's happened to the man he believes is his father. Jon is tempted to abandon the Night's Watch to help (a regular theme for Jon, considering he nearly did the same after the Red Wedding), but Aemon explains what happens when duty is abandoned.
"Love is the death of duty," Aemon says. "If the day should ever come when your lord father was forced to choose between honor on the one hand and those he loves on the other, what would he do? ... What is honor compared to a woman's love? And what is duty against the feel of a newborn son in your arms? Or a brother's smile?"
The inclination here is that Aemon is talking about himself; he later revealed in that scene to Jon that he was secretly a Targaryen and couldn't help his family when they were in need, even though he wanted to. But the same notion applies to Ned's decision to keep Jon and label him his bastard son for his safety.
It's one of the many ways in which Aemon mentored Jon before his death in season five. However, while he may have helped Jon repeatedly because he saw good leadership traits in him, it would be even more telling if Aemon was instructing someone who's technically his great great nephew. Moreover, Aemon's advice has consistently veered toward the symbolic — his "kill the boy Jon Snow and let the man be born" speech can be plainly interpreted as Jon's death and resurrection.
Sadly, whether or not the late Aemon knew about Jon's parentage will most likely never be addressed. However, at the very least, it adds another layer to Aemon's tutelage if he did knowingly help out his kin.
You can rewatch the season one scene between Aemon and Jon (back when Jon wasn't yet resurrected but was still very mopey) below.