I just wanted to clear up a misconception regarding Sheen that has really been bugging me.
First of all, the stats:
Sheen (1260 g)
Sapphire Crystal (400 g) + Amplifying Tome (435 g) + 425 g
UNIQUE Passive: Every time you use an ability, your next physical attack deals 100% extra base damage.
The mana and the AP are straightforward. What makes Sheen interesting is the passive. This is where a couple misconceptions are found.
1. Sheen only doubles base damage. If I have 100 base damage just from leveling and then a fully charged Bloodthirster for +100 damage, an attack with Sheen active will do 100*2 + 100 = 300 damage NOT (100+100)*2 = 400 damage.
2. Sheen damage cannot crit. If I have 100 base damage and no damage items, a crit with Sheen active will do 100*2 +100 = 300 damage NOT (100*2)*2 = 400 damage.
So for the big whammy, if I have 100 base damage and 100 Bloodthirster damage, a crit with Sheen active will do (100+100)*2 +100 = 500 damage NOT (100+100)*2*2 = 800 damage.
Let's look at a really extreme example. Blitzcrank has the aforementioned 100 base damage and 100 Bloodthirster damage. He uses Power Fist which activates the Sheen buff. On his next attack, he crits for (100+100)*2+(100+100)+100=700 damage NOT (100+100)*2*2*2=1600 damage.
If you want to see the interaction between Sheen, +damage, and crits in action, I suggest playing a practice game with no masteries and runes as pirate. A level 16 pirate has 100 base damage, and Parrrley level 5 gives 100 bonus damage, perfect for following along with the example. Pick up a sheen and some crit % items and you can easily see how everything works by shooting minions (they don't have any armor).
I hope this helped clear up some misconceptions surrounding Sheen and helps you theorycraft some awesome builds!