My maternal great-grandparents all came over from Norway in the late 19th and early 20th century. The latest was one of my great-grandmothers in 1902.
Counting both Canada and the US, my father’s side has been in North America for a lot longer. After doing some research, I can pinpoint that the most recent immigrants came over from Ireland sometime in the early to mid part of the 19th century.
I’m gen 15, since 1623, and there are 1 or 2 generations after me now.
Dad’s side, three. Or four, depending on whether great grandpa who was born in Italy counts. Or maybe five, since I have a son. Mom’s side? Who knows. An aunt who does genealogy traced a relative back to the American Revolution, but I’m not sure before that. Fun fact: apparently one of my ancestors was the guy who wrote “Home On the Range”.
Mayflower, including this guy.
I’m in the same boat- not the Mayflower, the Elizabeth;), 1630. There was a Moffatt who died at Ticonderoga, of dysentery. I notice they took their place names with them…a trail of Duxburies, culminating in Vermont, and many are still there. They were Cromwellian soldiers and yeomen, and tough as boot leather. My father’s side is more recent, but came from Ireland and Bavaria before the 20th century, the Irish during The Great Dying caused by the potato blight.
I have a cousin who is a Mayflower descendant. Her father is descended from someone on the Mayflower, but he converted to Judaism. His grandparents were really into the Mayflower/DAR stuff, and his grandmother wasn’t too happy, but his parents couldn’t have cared less. So she can be a member of the Mayflower Descendants Society (her grandmother gave her some kind of plaque to that effect once), and B’nai B’rith. I’ll bet not too many people can do that.
My ancestor Ephraim Blaine settles here in 1647. I have no idea what gen. I am considered to be. I do however know our family has done its part and then some. I take pride in the what so many of my ancestors have done for this, our, my country.
One generation. My parents came in the early 50s
I’ve done a bit of research since then. My maternal grandfather’s people were Famine Era immigrants on his father’s side; his mother who died young was Irish-born.
My “Scotch-Irish” Grandma had a grandfather born in New York City–also of Famine Era immigrants. But some of her people did come over from the more northern part of Ireland–before the Revolution. Possibly some Welsh & French in that lineage, too.
12 generations on one side, at least that on another. One ancestor born in Amsterdam in 1599, well established in New York in 1638. Another, of the same generation, described as an “early Connecticut settler” with no indication of whether he was first-generation or native-born.
All four of my grandparents were born in Ireland and moved to New York in the 1920s.
So they shambled into America in their hunt for braaaiinss?
My mothers family came in the 1680s, my fathers in the 1750s. I have no idea how many generations this is, but it surprised me how many… 15?
After World War II, my mother was a German war bride of an American soldier, and my father and his family were displaced persons (DP’s) from Latvia.
In Europe, I can trace my family back to the late 1700’s in Latvia, and to around 1640 in Germany.
My mother’s side goes back to the early Massachusetts settlers, as we’re direct descendants of Anne Lucy Howard (the same Howard family as the Taft politicos).
We have never been able to find a trace of my father’s family. What he told us (parents dead, no siblings) has never been confirmed.
A couple of strands’ worth of ancestry go back 10 generations. (That would be 8 from me; I have a niece who has a baby, so starting from the baby’s perspective and looking back over his tiny shoulders, that’s 10 gens).
I come from a long illustrious line of rural redneck farmers 
My paternal grandmother’s parents immigrated to the U.S. from Ireland, sometime in the 1890s; she was born here in 1902.
My maternal grandfather’s family traces back to immigration to English colonies in what would become the U.S., sometime in the 17th century.
My two other grandparents’ lines arrived in the U.S. sometime between those two. 
'Sup, cousin? (Actually I haven’t completely confirmed this yet; there’s one birth record in Connecticut I haven’t dug up yet that would provide the smoking gun.)
I have ancestors all over the timeline, from the Mayflower (John Alden and Priscilla Mullins on Dad’s side, potentially William Brewster on Mom’s) through early Boston and Dorchester in the 1630s (fellow named Samuel Pelton) and upstate New York in the mid-17th century (Emricks, some of whom got kicked out and went to Canada during the Revolution, then snuck back in 60 years later) to Irish and English immigrants in the 1850s to Swedish immigrants in the 1870s and 1890s. And there are some branches of the tree I haven’t yet far enough back to see where they came from, though it appears to be more of the seventeenth- and eighteenth-century settler variety.
So I guess it depends on which ancestor you’re looking at. Anywhere from 4 on up.
Not that long, the late 1840s for both my mom’s side and my dad’s side of the family.
I don’t really know, but pretty far. Some of my ancestors were colonists in North America a century before the formation of the USA.
I honestly wonder what the nationwide survey of citizens and permanent residents would be on this topic, framed in the context of “what percent of Americans have at least one grandparent not born in the US?”. If someone had a link to some kind of census information, that’d be great.
To answer the question in the OP: I’m the first generation in my own immediate family born in the US.
Moreover, the majority of kids I grew up with - not just relatives or family friends, but schoolmates and neighbors and kids at school - had at least one grandparent who was born elsewhere. At least half had at least one such parent (not just grandparent). And this isn’t because I mostly hung out with my own ethnic group - far from that, actually.
I would guess the majority of people I knew all of whose known ancestors had been here in this country for more than 3 generations were black (African-Americans), for obvious reasons. And even then, many of my “black” friends had some Latin American or European (from Europe) born ancestor within 3 generations.
At some point, later than you might think for someone rather highly educated, I realized this was atypical for the United States as a whole. Like, when I was maybe 16 years old? Comes from being from NYC, I suppose.
But I don’t know how atypical this is, or how distributed this is across the country.