Besides British immigration, did the thirteen colonies see a lot of immigration from other European states?
UPDATE: Aaron Fogleman estimates that 585,000 people "immigrated" (many involuntarily) to the 13 colonies between 1700-1775. If they all survived until the time of the Revolution, then 24% of the population at the time of the Revolution would be foreign born (585,000/2,400,000 = .244). Of course this is an absurd assumption, so treat this estimate as an extreme upper bound.
Here is the breakdown by ethnicity:
Source: Fogleman 1992. "Migrations to the Thirteen British North American Colonies, 1700-1775: New Estimates." Journal of Interdisciplinary History.
New England: New England was the most ethnically homogenous region and had the fewest new immigrants in 1776. New England always had the highest rate of family immigration, giving it a high birthrate. It had the lowest death rate, due to a healthy climate. Its low rates of non-English immigration are due to its cultural intolerance. However, far northern New England had a French splash from Acadia.
The South: The tidewater South was more ethnically diverse than New England, but that's mostly due to the presence of slaves. It initially had a lower birthrate, due in part to the immigration of single men looking to make their fortunes. Southern states had the highest death rates, due to malaria, yellow fever, and (some argue) physical violence. Still, these colonies had been settled early enough that they were still dominated by established families and third-plus generation yeomen.
Slaves, of course, succumbed to overwork--especially in South Carolina, which learned a crueler version of plantation management from the Caribbean colonies. In the upper south, slaves had been in the colonies long enough that we might speak of "African-Americans," whereas in the deep south the slave population had to be "replenished" at such a rate that we might still speak of "Africans" (which is why there are more vestiges of African culture there than elsewhere).
Appalachia: Little ethnic diversity here, as it was largely settled by Scots-Irish and others from northern England in the 18th century. At the time of the Revolution, this region would have been dominated by 1st to 3rd generation Americans.
Middle States: This is where all the ethnic diversity was. The Dutch were prominent in New York (around a quarter of the population), especially along the Hudson River Valley. Most Jews in colonial America were of Dutch Sephardic descent. "New Sweden" (running along the Delaware River Valley) initially had a sizable contingent of Swedes and Finns, though this was of diminished importance by 1776. Perhaps the most important non-English group in the Middle States was the German Palatines, who made up about a third of Pennsylvanians. Finally, maybe around 5% of the Middle States population were black slaves. As in the upper South, they were likely to have been descendants of several generations of slaves. Most of the immigration to this region had occurred by the early 18th century, so the Middle States too were dominated by third-plus generation Americans.
As a final aside, if you believe that the origins of the American Revolution were largely ideological, then it's no surprise that the Revolution occurred at a time when such a large percentage of the population was third-generation or older. The development of institutions and philosophies of governance incompatible with those in Britain doesn't happen over night, but over multiple generations.