I usually go with the default difficulty setting more to enjoy the story & the game itself for what it is.
If I really liked the game, I would start another run. Sometimes for fun, I would use cheats and what not and sometimes to play on hardest possible difficulty and see how that would go.
I really loved Metro and they had quite a nice difficulty settings which forced you to think about ammo conservation, different approaches to certain missions and such. That is what makes it even more fun.
Another game that I enjoyed playing on higher difficulty was GTA V. Multiplayer has some really nice heists and other than earning more money by doing harder runs, it was really fun to go through certain heists on hard difficulty. Even worse, you could have spent more than an hour to get close to ending it, fail and have to start everything from the start if your team wasn't really well coordinated.
Me and my friends actually found a really nice way to deal with the last heist that not many people have thought of at that time. The last heist, bank robbery. After you rob the bank you have to fight your way to the bikes & ride em to your freedom. With random people in your team it can be quite hard to communicate, but when you play with a team of friends and you are all voice chatting it benefits a lot. We would just wait inside the bank for the first wave of cops and swat to show up, 3 players would provide covering fire while the 4th member would get an RPG or grenade launcher and give them hell. When you go outside, do the same. Take down the chopper and incoming police cars with explosive weapons and let your mates cover you. Once you eventually get to the bikes, drive those to the beach. Sounds crazy? Perhaps it might, but on certain part of the beach you can snatch jet-skis and leave the annoying part of ridding the bike towards the safety and enjoy the more easier way of avoiding cops. It was such a nice, yet simple discovery which made the heist way easier. Hope this might help some other GTA V players