I'm going to assume you mean the highest of each stat for each different type. Not counting Ubers :3.
Normal
HP - Blissey (714)
Attack - Slaking (460)
Defense - Miltank (339)
Special Attack - Porygon-Z (405)
Special Defense - Blissey (405)
Speed - Meloetta Pirouette Form (390), if not counting this, Swellow (383)
Fighting
HP - Hariyama (492)
Attack - Conkeldurr (416)
Defense - Cobalion (392)
Special Attack - Keldeo (392)
Special Defense - Virizion (392)
Speed - Meloetta Pirouette Form (390), if not counting this, 4 Musketeers + Infernape (346)
Fire
HP - Entei (434)
Attack - Darmanitan (416)
Defense - Torkoal (416)
Special Attack - Chandelure (427)
Special Defense - Flareon (350)
Speed - Infernape (346)
Flying
HP - Drifblim (504)
Attack - Archeops (416)
Defense - Skarmory (416)
Special Attack - Zapdos/Moltres (383)
Special Defense - Mantine (416)
Speed - Ninjask (460)
Water
HP - Wailord (544)
Attack - Kingler (394)
Defense - Cloyster (504)
Special Attack - Keldeo (392)
Special Defense - Mantine (416)
Speed - Starmie (361)
Grass
HP - Amoonguss (432)
Attack - Breloom (394)
Defense - Ferrothorn (397)
Special Attack - Exeggutor/Roserade (383)
Special Defense - Virizion (392)
Speed - Sceptile (372)
Poison
HP - Amoonguss (432)
Attack - Toxicroak (342)
Defense - Weezing (372)
Special Attack - Gengar (394)
Special Defense - Tentacruel (372)
Speed - Crobat (394)
Electric
HP - Lanturn (454)
Attack - Electivire (379)
Defense - Magnezone (361)
Special Attack - Magnezone (394)
Special Defense - All Rotom Forms, discounting original (344)
Speed - Electrode (416)
Ground
HP - Rhyperior (434)
Attack - Rhyperior (416)
Defense - Steelix (548)
Special Attack - Camerupt (339)
Special Defense - Claydol (372)
Speed - Dugtrio (372)
Psychic
HP - Wobbufett (584)
Attack - Metagross (405)
Defense - Metagross (394)
Special Attack - Darmanitan Zen Mode (416), if not counting this Alakazam (405)
Special Defense - Bronzong (364)
Speed - Alakazam (372)
Rock
HP - Rhypherior (434)
Attack - Rampardos (471)
Defense - Shuckle (614)
Special Attack - Omastar (361)
Special Defense - Shuckle (614)
Speed - Aerodactyl (394)
Ice
HP - Lapras (464)
Attack - Mamoswine (394)
Defense - Cloyster (504)
Special Attack - Glaceon (394)
Special Defense - Cryogonal (405)
Speed - Weavile (383)
Bug
HP - Yanmega (376)
Attack - Escavalier (405)
Defense - Shuckle (614)
Special Attack - Volcarona (405)
Special Defense - Shuckle (614)
Speed - Ninjask (460)
Dragon
HP - Garchomp (420)
Attack - Haxorus (432)
Defense - Shelgon (328)
Special Attack - Hydreigon (383)
Special Defense - Altaria (339)
Speed - Garchomp (333)
Ghost
HP - Drifblim (504)
Attack - Golurk (381)
Defense - Cofagrigus (427)
Special Attack - Chandelure (427)
Special Defense - Dusknoir (405)
Speed - Gengar (350)
Dark
(Credit to MysticMaxwell btw)
HP- Mandibuzz (424)
Attack - Tyranitar (403)
Defense - Scrafty (361)
Special Attack - Hydreigon (383)
Special Defense - Umbreon (394)
Speed - Weavile (383)
Steel
HP - Excadrill (424)
Attack - Metagross/Excadrill/Escavalier (405)
Defense - Steelix (548)
Special Attack - Magnezone/Heatran (394)
Special Defense - Probopass (438)
Speed - Durant (348)
Fairy
No one knows :3
Finally done! Feel free to correct if I stuffed up something or missed something
Adding in Gen 6 later