Spoilers for all books and TV seasons up to the end of A Dance With Dragons/Season 5.
Known Targaryens who are still alive:
Daenerys Targaryen,
Brynden Rivers/Bloodraven,
Stannis Baratheon (in the books),
Shireen Baratheon (in the books)
Known Targaryens who have died during the series:
Viserys Targaryen,
Maester Aemon,
Robert Baratheon,
Renly Baratheon,
Stannis Baratheon (in the TV series),
Shireen Baratheon (in the TV series)
Possible/Suspected Targaryens:
Aegon VI (claims to be Rhaegar's son, more likely Illyrio's son),
Tyrion Lannister (Aerys was fond of his mother),
Cersei & Jaime Lannister (ditto),
Jon Snow (R + L = J),
any Blackfyres in the female line of descent who are still kicking around in Essos,
any descendants of Aerion Brightflame who are still kicking around in Essos,
Illyrio (maybe a Blackfyre),
Varys (maybe a Brightflame),
Joffrey, Tommen & Myrcella Baratheon (believed to be Robert Baratheon's children and thus have Targaryen ancestry, but we know they're not)
My Personal Crackpot Theory:
Littlefinger's grandmother (who we know came from Essos) was a Blackfyre, and he has Targaryen blood and will end up as King. He is already quite like Henry Tudor, who ended up as Henry VII after winning the Wars of the Roses, and Henry Tudor's claim to the throne was because he was descended from the Plantagenets in an illegitimate female line.
"two drops [of dragon blood]"in him. – Möoz Feb 09 '16 at 21:02