I'm not sure how RussSeaton defines "real mountains,"
I don't consider a climb a mountain unless its 6-7-8 miles of climbing up. Constant or varied grade. Switchbacks too. 1-2-3 miles and its still a hill. Maybe a loooong hill, but not a mountain. The 4-5 mile length I guess you could put in either category depending on how vigorous you were that day. I also think of mountains as having a pass at the top. Usually a named pass. Hills usually don't have pass names and elevation signs at the summit.
All of this is OBE since OP is riding west to east, but...
This incredibly restrictive definition of a "mountain" should probably be adapted for local variances. In the southern Appalachians, we call them "gaps" instead of "passes." Further north, the same thing may be a "notch."
Even so, the climb from the Clinch River up to Hayters Gap in Clinch Mountain, two weeks from the east coast, fits Russ' restrictive definition. Except maybe for the sign; I don't remember one. But then again, there wasn't a sign when I rode across Togwotee Pass (second highest pass on the TransAm). Do people riding east get to count it as a mountain after 15 miles of climbing if there's no sign?
Personally, I thought the mountains in western Virginia, eastern Kentucky, and central Missouri (Ozarks) were the toughest on the TransAm. By comparison, the passes in Colorado, Wyoming, and Montana were easy grades. Yes, they were long, but the grades were much easier. Fortunately, the OP will be ready for the tough stuff by the time he gets there coming east.
Correction: west bound (going north) you cross the North Fork of the Holston before climbing Clinch Mtn. The Clinch River is north of Clinch Mtn. Still a good climb, but not as steep eastbound.