Madonna's albums are all untitled.
`` Like a Virgin '' is a song recorded by American singer Madonna for her second studio album Like a Virgin ( 1984 ) . Madonna (entertainer) . Throughout her career , Madonna has written and produced most of her songs , with many of them reaching number one on the record charts , including `` Like a Virgin '' , `` Papa Do n't Preach '' , `` Like a Prayer '' , `` Vogue '' , `` Take a Bow '' , `` Frozen '' , `` Music '' , `` Hung Up '' , and `` 4 Minutes '' .