Round and Round the Garden
The version given by The Oxford Dictionary of Nursery Rhymes is:
Round and round the garden
Like a teddy bear;
One step, two step,
Tickle you under there!
This rhyme is probably fairly recent.
The rhyme was first collected in Britain in the late 1940s. Since teddy bears were not common before the twentieth century it is likely fairly recent in its current form. Iona and Peter Opie suggest that it is probably a version of an older rhyme, "Round about there":
Round about there
Sat a little hare,
The bow-wows came and chased him
Right up there!