Answer: 147th New York and the 84th New York (who preferred to be known as 14th Brooklyn)
Source: Stephen W. Sears Gettysburg page 364
Edit - I don't have a copy of Sears' book, so I don't know exactly what it says on page 364.
Michael Martin's book, cited as a source by WJC, says the 147th NY and 84th NY were the only two regiments - from either army - to be engaged in "major combat" on all three days. Maybe Sears' book also used the "major combat" terminology. The question, however, only asked about regiments that "fought" on all three days.
I would think that you could "fight" without necessarily engaging in "major combat." And our players found sources indicating that a good many more than two regiments "fought" on all three days. These include the 45th and 76th NY, the 6th and 7th WI, the 143rd and 150th PA, the 22nd MA, the 16th ME, and the 55th OH. I am considering all of these to be acceptable answers.
Sources have also been submitted to indicate that the 147th NY was known as either the "Ploughboys" or the "Oswego Regiment," and the 84th NY / 14th NY / 14th Brooklyn was also known as the "Red Legged Devils." I will accept those names as well.
hoosier