I know that Boruto has at least two Rasengan, but I don't know exactly how many he has.
There are a total of 9 Rasengans, and all can only be attained by combining the power of the tailed beasts. As of right now, Boruto has 3 types that he can use with the help of his Karma (Which was made to Boruto by Koji Kashin a Jiraya clone), the three types are Wind Style Rasengan, High Compression Rasengan & Vanishing Rasengan.
Wind Style Rasengan was taught to him through an explanation by Kakashi then the Wind Style Rasengan evolved into the High Compression Rasengan which Boruto can only use once in extreme cases.
Vanishing Rasengan Was created by Boruto as a result of his chakra nature.