Honestly, I think this is a big sign that it's time to start the process of potty training. Probably not the solution you were expecting to hear, I know, but he's not too young at all and potty training can be really enjoyable and fun for BOTH mom and toddler. :)
Seriously, you can spend more $$$ and go up a size in diapers, buy plastic pants, use 2 diapers at a time, or find & buy diaper doublers... but why put all your time, effort, and $$ into ALL THAT HASSLE when you are just prolonging the inevitable (potty training)??? Or, you can do now what you'll eventually need to do anyway, and that's help him to reconnect his body sensations with the outcome (pee/poop) and use the potty instead of the diaper.
I think it's a bit illogical and silly that in our culture, we don't wait for a child to tell us he's hungry before we feed him, and we don't wait for a child to tell us he's sleepy before we help him to bed, but for some reason these days (and this is only in the last 75 years, and only in western countries) we're supposed to just ignore elimination signals because we're supposed to "wait until they tell you they're ready!" C'mon, a kid who is peeing all the way through their diapers in the morning, or grunting and red-faced when they poop in the diaper after lunch might not be able to articulate to you IN WORDS "hey, I don't want to be messing all over myself" --> but trust me, they probably don't want to be doing that and need help from you to get to the toilet! (Would you want to go on yourself in a diaper?!? Didn't think so.) ;)
So, that's why I'm suggesting this "thinking-outside-of-the-box solution" to your leaky diaper dilemma. Dive on in and start the potty training process - you can do it!
An interesting fact for you about why your son is probably leaking out of his diapers: Humans in general typically pee LESS while they sleep (a hormone called vasopressin increases while we sleep, reducing our need to pee while asleep). So, by the time a baby or toddler wakes after a several-hour-stretch or all-night, it's typically due to *bladder pressure* and they have a huge pee, soaking the diaper completely through. You don't notice it when they're younger because the amount that they pee is smaller. But at age two, it's a lot more pee!
Again, your son is *definitely* in range for potty training, and expect it to be a slow process, like starting solids was a gradual process. Sure, you can wait, though for most I think it just makes potty training more of a struggle.
So, my suggestion really does go back to the whole leaking through the diapers issue! So you know your son drenches his diaper in the morning pretty reliably --> by the time you get to him, is the diaper warm or cold? If it's NOT cold, you can be pretty sure he peed RECENTLY in the diaper. So instead of spending your time washing and changing sheets, I suggest you refocus that effort and time on simply listening for his first waking sounds in the morning, at which point you would go into his room, gently remove his diaper, and hold him supportedly *over* a little plastic potty (if you sit him on it, make sure the plastic isn't too cold for him). Talk to him, "instead of peeing in your diaper, I'm going to help you pee in the potty, sweetie!" and make a "pssss pssss" sound. Keep communicating with him about what you're doing, about what's happening ("look sweetie, your pee pee is coming out and going into the potty - what a big boy!"). It may take 30 seconds or so for him to relax into being held over the potty, and you'll feel him release and start to pee. Have fun, make it NO STRESS, work as a team. If you do this every day for a week, you'll likely:
1) catch that big, first pee of the day -- helping your son regain awareness of his body signals that are telling him "bladder pressure, pee!!!" (Hint: same for right as he wakes up from naps!)
2) avoid changing and washing so many sheets!
3) save $$ in diapers
4) waste less (great for the environment)
4) avoid having a 3 yr old (or 4 or 5 or 6 yr old) who doesn't want to poop anywhere but in their diaper
5) have fun working as a team and increasing your connection with your son as you go through this new, cool learning adventure together
So... happy pottying! Below I am ending w/ some great resources, if you want to explore the idea of potty training him now instead of waiting:
Part-Time Potty Training Tips for All Ages
http://www.viviente.com/2005/10/how_to_do_parttime_pottyt...
Early Start Potty Training by Linda Sonna
http://www.drsonna.org/ptresources.htm
Diaper-Free by Ingrid Bauer
http://www.natural-wisdom.com/
Diaper-Free Before 3 by Dr. Jill Lekovic
www.diaperfreebefore3.com/