Nothing really happens except you can't level it up anymore. It's really easy to end up with Connections: Society 90 if you max out your Persuasive, since basically everything in the Shuttered Palace gives you Connections: Society. It's the only one that's that high. I have The Church at 44 and the Constables at 55 and everything else is mostly in the 20s or lower. [note that I spent some of the 90 Fate I got from that weekend that got erased to unlock the Soul Trade, and if you take the path of restoring souls to people, the Church, Society, and Constables are basically the connections you end up getting a lot of once you reach Shepherd of Souls 5]

I'm currently grinding my Dangerous in lower levels with my Ridiculous Hat + Talkative Rattus Faber because I need to max out my Dangerous to finish up the Nemesis plot over in the Tomb Colony of Venderbight, but I've already got a Running Battle... going over there, so I can't duel because I'll forget I have to save it and will waste it killing Feducci or something.