I've only had one league since I started playing OOTP in 2007. I used real teams and logos with fantasy players. I just finished 2047 and have had a blast!
I started as the gm of St. Louis and only recently retired at the age of 90. (See another thread about resigning as gm in OOTP 17 forum.) I really lucked out in the beginning when my starting shortstop was an 18 year old kid that could run, hit, and play defense. He hit around .300 every year and usually led the league in caught stealing, but he also stole around 60 bases too. And man, could he suck up some grounders in the field! When he got to be about 35, his skills started to drop fast, but he was still under contract for another couple years. I couldn't bear to trade him. I needed the roster spot, but I kept him on the bench anyway. In his final game he got 3 hits, and we were going to the Series! He retired with a career .300 average and was elected to the Hall of Fame.
In our second season in 2008, we signed a free agent outfielder that ended up hitting .400. When he got injured in late August, I almost had a heart attack. He came back off the DL in late September and still needed a few more Plate Appearances to qualify for the batting championship. He stayed healthy, got the PAs he needed, and we won our first World Series!
There are many more stories like that. I even have a soft spot for certain players that either played for me and left in free agency, or rivals that were great and earned my respect.
I put the game away every 6 months or so. When I come back, I remember all the names of the key players and get sucked right back in.
Don't know if I'll ever bother to start a new league. This one's too dear to my heart.