On my mother's side, I can go back 4 generations. She and most of her ancestors are British, but I have a Goanese great-great-grandmother.
On my father's side, I can go back 3 generations - that's the generation that immigrated from Poland.
One of my MIL's hobbies is genealogy, and she's traced both her family and her husband's family pretty far back. She has also tried to trace my father's genealogy - she is on a quest to find some Jewish ancestors for me, but I'm afraid she'll have to go back nearly 2000 years for that