I did math.
I have a habit of finding the most of whatever at level 1. Here's what I found:
Max Rune Attack Speed: ((Quints=13.5), (Marks=15.3), (Glyphs=5.76), (Seals=6.84)) = 41.4%.
Items at Level 1: Dagger gives 12%.
Masteries: Fury gives 4%.
When playing Olaf: At 99% Health (since you can't have 100% or you'll be dead) you will have 99%.
Nidalee 'E' Ability: Gives 20% at level 1.
Ezreal 'W' Ability: Gives 20% at level 1.
Warwick 'W' Ability: Gives 40% at level 1.
Jarvan IV 'E' Ability: Gives 10% at level 1.
The total amount of attack speed at level 1 is 246.4%.
If you can do math as well and find a different answer then good for you, I am simply just trying to give a rough number of what is achievable.