With just the chords C, G7 and F you can play a great number of songs, because basic song and tune structure calls for these alone.
But by adding Am (A minor), and possibly Dm and/or C7, (all easy chords on the ukulele) the structure of the songs becomes more interesting, and - you can play a lot more of them!
Waltzing Matilda C G7 F Am
Drunken Sailor C G7 F Am Dm
Fulsom Prison Blues C G7 F C7
Jackson C G7 F C7
Rock Around the Clock C G7 F C7
I Can't Stop Loving You C G7 F C7
I Wish it Would Rain C G7 F Am
This Land is Your Land C G7 F Am
Dirty Old Town C G7 F Am
If I had a Hammer C G7 F Am
Doo Wah Diddy Diddy C G7 F Am
Eve of Destruction C G7 F Am
Please feel free to suggest additions to this list via the Comments!
Just a note on minor chords, for those who aren't familiar with them. Without going into music theory, you can recognize a minor chord because it sounds "sad" whereas a major chord (a chord that just has a letter as a name) sounds bright and cheerful. Simple, really. So the occasional minor chord adds a bit of shift between light and shade to the music. More interesting!