Two Block Jump?

There's no way to jump any higher in a vanilla single player world.

From the Minecraft Wiki page on Transportation:

Using any method of elevating yourself that relies purely on jumping, including climbing a slope without stairs and pillar jumping, you cannot rise faster than about 2.0 m/s.

Sprinting allows you to jump further horizontally, but not vertically. From the Minecraft Wiki page on sprinting:

It is possible to jump across four blocks horizontally (although difficult), instead of the usual two blocks by sprinting, however the jump height is no higher than normal.

Actually, you if you can manage to set yourself on fire or poison yourself, if you jump at the exact time you take damage, you can jump 2 blocks. I've done it many times before, but I think the best way to prove this is by jumping up a cactus. Try it yourself and see! I am also making a video about it, I'll add the link when done.

I believe that it’s possible to vertically make a 2 block jump. I was watching a video called last to stop running wins $1,000. The jump is somewhere around 5:15. He uses cactus blocks to make himself jump higher. Here’s the link: