ORIGINAL: Grotius
If you impose a few modest constraints on yourself, the AI can give you a good game even on Historical or Hard level. I am on Hard, playing as the Allies, June 1942. The AI has the Phillipines, DEI, Malaya, Burma. I didn't put the US CVs in the DEI, or move a USMC division into Batavia, or strip all of India to defend Singapore, or anything like that. I limit replacements in areas that seem to me isolated from replacements. And I've been trying to follow a course parallel to that followed by Allied commanders in the war. So instead of piling all my marines into an attack on Eniwetok or other CentPac bases, I'm building up in the South Pacific, moving my CVs there, etc.
I also never -- EVER -- save and reload. That's a huge "cheat" in favor of humans. If part of a ground unit doesn't load, I plan for that and send a follow-up transport TF with the remaining bits and pieces to follow it. If I lose a CV battle because I forget to put up CAP, so be it.
Also, I'd like to try playing against the Allied AI on Very Hard in a 1944 or 1945 scenario.
Sure, PBEM will always give you a more intelligent opponent. But AI has its advantages. Because it tends to play within a historical framework, you you can get a war more like the real war. It's fast; you can actually play the whole war. I can realistically hope to finish my current game in the current calendar year. AI doesn't quit. And it doesn't care if you want to play 10 turns a night or take a few days off.
This is the way I do it too. I prefer PBEM but sometimes you want to play and PBEM turns are not there [:D] I keep a game running vs the AI where I play with similar 'House Rules' as Grotius.
I'm Japan in mine, so I try and follow the historic landing schedule. I break up KB in 2-3 TF's and rest and refit the CV's between operations. I rest my air units when moral drops or fatigue rises. Rest and prep my ground units before untertaking operations. You get the idea.