I know that this answer is extremely late, but there are some newer ways to get the highest stats possible in Pokemon. All of these have been tested using Pokemon Showdown. Additionally, the special defense and speed methods were available to you during the time of the post, so the other answers are incorrect.
Disclaimer: I will say "a move that doesn't affect" a certain Pokemon multiple times throughout this answer. This just means that the move does not affect that Pokemon's stats or HP. You may use many moves that boost your stats such as Swords Dance or Calm Mind. This move may also be Protect (In some cases) and Substitute (In some cases).
Highest HP Possible: A level 100 Blissey with at least 252 EVs and 31 IVs in HP, giving it a total of 714 HP
Highest Attack Possible and Highest Stat Possible:
Requirements:
1. A level 100 Shuckle with at least 252 EVs and 31 IVs in defense, with a defense boosting nature
2. Shuckle must have Power Trick and Shell Smash
3. A Bunnelby or Diggersby with Huge Power and U-turn
4. Other team must have a Pokemon with Skill Swap that is faster than Bunnelby or Diggersby and knows 1 move that doesn't affect Shuckle
5. Other team must have another Pokemon with Trick and must hold a Choice Band
Achieve By:
1. Sending out Bunnelby or Diggersby and the other team must send out the Pokemon with Skill Swap
2. Other Pokemon uses Skill Swap while Bunnelby or Diggersby uses U-turn and sends out Shuckle
3. Other Pokemon uses Skill Swap again while Shuckle uses Power Trick
4. Other Pokemon uses the move that doesn't affect Shuckle (Not Trick) while Shuckle uses Shell Smash
5. Other Pokemon switches into the Pokemon that knows Trick while Shuckle uses Shell Smash
6. Other Pokemon that knows Trick uses Trick while Shuckle uses Shell Smash to give Shuckle the Choice Band
7. Shuckle now has 7,368 attack
Highest Defense Possible:
Requirements:
1. A level 100 Shuckle with at least 252 EVs and 31 IVs in defense, with a defense boosting nature
2. Shuckle cannot have Contrary
3. Shuckle must have Defense Curl or Withdraw
4. Other team must have Alolan Persian with Fur Coat, Assist, and 1 move that doesn't affect Shuckle
5. Other team must have only 1 other Pokemon and it can only know Entrainment or Skill Swap
Achieve By:
1. Sending out Shuckle and Alolan Persian
2. Alolan Persian uses Assist, which can only be turned into Entrainment or Skill Swap to give Shuckle Fur Coat ability while Shuckle uses Defense Curl or Withdraw
3. Alolan Persian uses the other move that doesn't affect Shuckle while Shuckle uses Defense Curl or Withdraw
4. Repeat step 3 four more times
5. Shuckle now has 4912 defense
Highest Special Attack Possible:
Requirements:
1. A level 100 Deoxys Attack with at least 252 EVs and 31 IVs in special attack with a special attack boosting nature
2. Deoxys Attack must know Nasty Plot and hold a Choice Specs
Achieve By:
1. Sending out Deoxys Attack
2. Deoxys Attack uses Nasty Plot
3. Repeat Step 1 two times
4. Deoxys Attack now has 3,024 special attack
Highest Special Defense Possible:
Requirements:
1. A level 100 Shuckle with at least 252 EVs and 31 IVs in special defense, with a special defense boosting nature
2. Shuckle must know at least 1 move that is an attacking move
3. Shuckle must hold an Assault Vest
4. Another Pokemon on the opposing team that knows Metal Sound
Achieve By:
1. Sending out Shuckle and the Pokemon that knows Metal Sound and is on the opposing team
2. Pokemon that knows Metal Sound uses Metal Sound
3. Repeat step 2 twice
4. Shuckle now has 3,684 special defense
Highest Speed Possible:
Requirements:
1. A level 100 Deoxys Speed with at least 252 EVs and 31 IVs in speed, with a speed boosting nature
2. Deoxys Speed must know Fling and Agility and hold an item that can be flung using Fling
3. A Drifblim or a Drifloon with Unburden and Skill Swap and 1 other move that doesn't affect Deoxys Speed
Achieve By:
1. Sending out Deoxys Speed and Drifblim/Drifloon
2. Deoxys Speed uses Agility while Drifblim/Drifloon uses Skill Swap
3. Deoxys Speed uses Fling while Drifblim/Drifloon uses the move that doesn't affect Deoxys Speed
4. Deoxys Speed uses Agility while Drifblim/Drifloon uses the move that doesn't affect Deoxys Speed
5. Repeat step 4 once
6. Deoxys Speed now has 4,032 speed
I hope that this information is helpful or feeds your curiosity.