Okay…so REALLY this list is 5, but I made it 10 to fit the format. Forgive me?
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First, my honorable mentions to make it 10: Louisville-Florida State, USF-UCF, Appalachian State-Georgia Southern, Pitt-Virginia Tech, and Wisconsin-Iowa.
Let's take a look at Nebraska and Wisconsin. These teams are probably the two biggest power in the Big Ten West and likely will battle over a place in the Big Ten Championship Game for years to come, with apologies to Iowa. In 2011, Wisconsin beat the Cornhuskers 48-17. Even though Nebraska won by 3 points in the 2012 regular season, Wisconsin won by 39 in the 2012 Big Ten Championship and by 35 in 2014. But the last two meetings have been memorable. In 2015, Wisconsin kicker Rafael Gaglianone kicked the game-winning field goal with 4 seconds left and the Badgers won in OT last season in 2016. If Nebraska can get back to the "old" Nebraska, this could be the Big Ten West's version of Michigan-Ohio State. It is played every year for the Freedom Trophy.
Utah and Colorado are another blossoming rivalry. These two played every single year from 1903-1958 (minus 1909) but didn't play from 1963-2010. When the two joined the Pac-12 to form the Pac-12 in 2011, the rivalry–now known as the Rumble in the Rockies–began again as conference members. All 6 games have been by a touchdown or less. Utah has won 4 of the first 6, but the Buffs have done something Utah hasn't yet–represented the Pac-12 South in the Pac-12 Championship Game. I expect this game to continue to be competitive and go down to the wire.
Next up is Clemson and Louisville. This new rivalry has been very competitive its first few years. Louisville has never beaten Clemson, but the Tigers' 3 home wins have been by a combined 15 points. All three games have gone down to the wire and last season's game between the two Heisman Trophy finalists decided the ACC Atlantic division. The game in Louisville, KY this season could do the same.
The final two both involve Texas A&M. Alabama is such a perfect rival for the Aggies, but their games against LSU and Ole Miss have been much more competitive. LSU has won the last 6 against Texas A&M (including the 2011 Cotton Bowl). LSU came back from a 12-0 deficit in 2012 to win 24-19 and 2013 was a blow-out. A late interception sealed a Tigers' victory in 2014. The last two season have been double-digit victories but it won't always be that way. A&M recruits too good for that.
If you take out the 20-point Ole Miss victory in 2015, the series between the Rebels and Aggies has been close. Texas A&M has won twice by a combined 6 points and Ole Miss has won twice by a combined 16 points. These two teams have been pretty competitive on the field and both have entered seasons recently with national championship buzz. Depending on the continuation of recruiting, this game should stay fairly competitive.