>> My question is, how do you start a LARGE fire using very wet wood? <<
You can't - not really. Chris K. put you on the right track. Smaller fires are usually better anyway.
It pays to mind what species of wood you're using, too - doubly so when it seems like all the wood is wet. Red oaks are sponges, for example. (I'm not crazy about any oak in a camp fire in the first place - fine in a fireplace or stove, but...) But sometimes you just have to make do with whatever is available.
So having enough dry fixings for a couple of fires can be a real life-saver. Carry some fatwood, at least. And have some means of splitting available wood into smaller diameters. Following assumes that you have a stout locking folder, fixed blade knife, hatchet, or small ax:
If the only wood you can find is wet, split it down to pencil diameter - a good -sized pile say about enough to fill up the big cookpot in your patrol cookset. Then split another cookpot full in pieces ranging from about 1/2" to 1" in diameter. There is no sense trying to get going faster - split up all that stuff first, and keep it from getting any wetter than it already is - keep it out of the rain. If the ground is totally soaked, make a platform with small diameter logs (2" - 4" in diameter) and build the fire on that. Start your fire with your fatwood and add the small diameter split stuff loosely - build it up but don't choke it. When you have a solid blaze going with coals starting to build up, start mixing in the larger split stuff in with the remaining small stuff. Don't put anything damp on the fire unless it's at least split in 1/2 and preferably split in 1/4s or better - the less bark and the more splintery faces and edges the better.
Ash is my favorite fuel wood; osage orange is great if you can break it up (spits a lot, but what coals!) and dry mulberry is almost as good if most of the bark is off, hickory is good and walnut not bad - well, plenty of good wood around here - but you need to split your wood in really damp conditions. Learn what good firewoods are common in your area
If the rain is really pouring down, you'll have a tougher time and might want to shield the fire from the rain. It's pretty miserable under those conditions and I can think of very few reasons why anyone would need to stay out in the rain messing around with a campfire - getting out of the rain is more important.
A tarp and a backpacking stove has always been my preference in those conditions. All my scouts carry at least an Esbit stove in their ready packs and many of them have moved to small backpacking stoves instead. An Esbit stove, a canteen cup, and a foil windscreen cut out from a disposable aluminum roasting pan has cooked many a meal and brewed many a cup of hot beverage for my scouts - food for thought. They don't PLAN to use the Esbit stoves (well, not usually), but they beat the heck out of cold food and beverage under those conditions. We get the stoves for around $5 each, which includes a box of fuel (Stove is Esbit brand and fuel is generic - the real Esbit Tabs are larger and cost more.) Many hardware stores carry Weber solid fuel charcoal starter, which looks and burns amazingly just like Esbit fuel (hexamine)...
For a very lttle effort and even less cost you could make alcohol stoves. There are many designs to choose from, but they all work roughly as well, even the simplest ones. A small alcohol stove, 3 tent pins (or 40 penny pole barn nails), and a foil windscreen could help you out... the smallest kettle in your patrol cookset will hold enough water for 1-2 scouts to make a meal. (Supposedly a 2qt kettle, but it only holds about 1 quart with enough headroom to boil water without slopping over).
In your neck of the woods any cannister stove will work pretty much year-round without being finicky - there are some very inexpensive ones available from Coleman, Markill (and re-branded Markill), etc. More on that some other time if you're interested.
And of course there are the old standbys (white gas stoves). Most troops around here just use an ordinary 2 burner Coleman stove (white gas or propane) when canoeing... still need a tarp if the rain is coming down.
Hope that gives you a few ideas - be a good idea to practice this stuff before you need it. I'll be out camping with scouts every weekend this month and they will be using, at various times, all of the above... including campfires. Going to be a great month...
ADV/ASM Ayers
Crew & Troop 258
Illowa Council