The Beatles onstage at the Top Ten Club in Hamburg, during the spring of 1961. From left to right: Stu Sutcliffe, George Harrison, John Lennon, Paul McCartney (at the piano), and Tony Sheridan. You can just make out Pete Best's drums at the far right of the photo.
Well, the short answer is many. One source I've read described the very early Beatles as a living jukebox (remember those?). Apparently, the band played everything from early rock&roll and rockabilly, to show tunes and torch songs in an attempt to fill the long hours they were contracted to play during their stints in Hamburg. Lots of bands playing clubs now are lucky to know maybe 40-50 songs. Maybe. And if there is more than one group on the bill, a band might be expected to play far fewer than that seamlessly. But by all accounts, The Beatles had about two hundred songs in their communal bag of tricks at this stage. Pretty impressive when you think about it.
So, the catalog of songs played by The Beatles in their early years is far too large to list easily here. At least right now. I'm working on it. In the meantime, here's one example of a tune that the band surely played onstage in Spring 1961 during their second visit to Hamburg after George Harrison turned 18 in February of that year, and the band was able to return legally to Germany. Cry for a Shadow (aka, Beatle Bop), is a parody of various instrumental tunes by The Shadows (yep, Cliff Richards' backing band), who had a number British hits in the early 1960s. Here's the link: http://www.youtube.com/watch?v=l9FacSNRBHk. Blogger is being temperamental today, otherwise, I would have embedded a direct link to YouTube within this post. But anyway.
What's neat about Cry for a Shadow is that it is the only Lennon-Harrison composition out there, so far as I know. This song was recorded as part of the Bert Kaempfert-produced sessions, in which The Beatles backed singer Tony Sheridan, in May or June of 1961. The Beatles were allowed to commit two of their own tunes to tape at that time, Cry for a Shadow and a cover of Ain't She Sweet, which will appear here later. Another neat thing about both of these songs is that they give a pretty good idea of what The Beatles must have sounded like onstage at this point in their career. Listen for Paul McCartney and John Lennon howling in the background at various points in Cry for a Shadow. You also get a good sense of McCartney's sense of melody and rhythm, to say nothing of his emerging talent, on the bass here. Enjoy!