Heavy armor wise, if i invest perks into the heavy armor tree, which armor can i improve to hit to 564? I want to avoid alchemy and enchanting.

Im assuming youre still using Smithing? Probably Ebony and up. Nordic Carved might too, I got it up to around 370 with 3 Juggernaut perks and about Epic quality.

None

Pretty much all of them if you pick the appropriate perks.

I have hit the cap using only game available Enchanted Items (for Smithing) to temper the armor. However, you can reach the cap with just about any other material with the right Perks and Skills

See this for how to get there. Yes, some of it talks about Alchemy/Enchanting, but get past that to the Perk/Skill strategies

http://www.gamesas.com/topic/1313851-complete-character-design-freedom-damage-resist-caps-and-ridiculous-damage-thread-7/

Perks? Heavy Armor perks?

Never!

But without perks my Ancient Nord armor is pushing ~500 with no shield.

(Smithing 100; Fortify Smithing 168% potion; 5 x 30% Smithing enchantments.)

Looks so awesome and it's wonderfully light, too.

But I guess sure, you could avoid Alchemy and Enchanting and instead put perks into heavy armor...

Of course it's all how you want to RP it.

Though Alchemy and Enchanting are a huge (major) part of Skyrim for me personally.

Always interesting to read the thoughts of others...

Alchemy and Enchanting? Powerleveling the game?
Never!

Joking, my point is -play and let play. Or something. Everyone has his own playstyle, at least TES is not one of the games where everyone runs around with a single ultimate build and equipment depending on the race/class.

I think steel plate at legendary quality, with all the AR increasing perks, does the trick with no enchanting or alchemy. I know that it is very close, if it doesn't go just over the AR cap. So steel plate, orcish, ebony, nordic carved, stalhrim, daedric and dragonbone should all be suitable options for you.

I think you can throw in legendary iron and everything of that quality if you get the Ancient Knowledge perk, since it is bugged and give a 25% bonus to all armor(light and heavy), even though it was meant to only affect dwarven. Possible it works like the custom fitted perk, requiring armor, gauntlets, boots and helmet, but I'm not entirely sure on that point.

I think the ancient knowledge perk works regardless if all slots are covered. That seems to be the case with my characters that don't use helmets. How do people have 5 smithing slots, btw? I only have 4 slots that will take the enchant

Dunmer shoes from Dragonborn also count as a chest piece.

That would explain it. I don't have that DLC so no extra enchant slot glitch for me