I own both - I have an American standard strat (all single coil pickups and a floating trem bridge). I also have a Gibson Les Paul Faded Studio with humbuckers.
Owning a strat was a dream since I was a teenager. I didn't get one until I was around 30. That was all I played until I got a the LP in 2011.
It's all a matter of personal preference, physical makeup, and what type of music you are trying to play.
For me, the LP seems to be a better guitar. I'm of smaller stature (5' 4") and the LP's shorter scale suits me better. Comparatively, the strat feels like a bass to me. Though, there are tones you can get from a strat (especially with pickup switch positions 2 and 4) that are unique and sound very "stratty".
I'm happy to own both and am learning to appreciate the pluses an minuses of each.
I was playing .011 on my strat and recently switch back to .009s. I much prefer the more twangy sound of the 9's on the strat. The standard 10's on my LP suit me fine. I'm going to try a set of .010's on my strat to see how that feels. The 11's I found just sound too lifeless to me as my playing advances.
Could I only live with one? Yes...and it would be the LP. I would be really sad to have to give up my strat though...I would really not want to give it up.