When it comes to arcade games I just don't continue. Something about repetition appeals to me, to play until you get to that wall and keep on trying until you break through just to run into another. Granted I haven't 1cc'ed worth a damn but I get more joy out of the challenge rather than playing tourist.
With console games I tend to just place the game until is beaten utilizing whatever is given to do such whether is continues, save points, or whatever, once I beat the game I proceed for a no continue run and if possible a no death run. Mostly with NES games but I also try with other systems.
Alien Infestation for the DS, game is amazing, I liked it so much that after I beat it I played it again until I had a perfect run without loosing any of the soldiers, so some challenges are self imposed I guess. My favorite game Ninja Gaiden 3 NES only has 5 continues, first time I beat the game I did it utilizing all the continues, after that I went for the 1CC, when I get back into it again, once I get 1CC again I'll shoot for the no death run.
I wish I could be as diligent with all my games as I am with good action platformers and fighting games, I do my best with shmups but I'm mediocre at best, have yet to 1CC Do Donpachi but I'm at the last boss now, I mean a few months ago when I was playing it on a regular basis. Same for Donpachi, its just them patterns are nuts.
Games like Ninja Gaiden Black I can't 1cc but I like to do speed runs, I'm no champ but I can go somewhat.
At the moment I'm playing Dark Souls like there is no tomorrow, I have a file on each system and trucking on getting the job done, once I beat it I'll be working on speed runs, its a great game.
So in short I'm a bigger fan of the challenge over the experience if that makes any sense but I enjoy the experience much more when I get to that point by beating the system as opposed to being carried there.