Can I choose any items and have their effects stack?

Solution 1:

You can stack items, however any unique properties will only apply once. Some stats stack differently than others (math) but they ultimately do stack.

For example, Madred's Bloodrazor provides the following:

+30 Attack Damage
+40% Attack Speed
+25 Armor

UNIQUE - On hit, deals Magic Damage equal to 4% of the target's maximum Health.

Buying more than one of these will give you the additional attack damage, attack speed and armor, but not an additional unique effect (4% of target's health per hit).

You can have more than one unique effect, but you may only have one of each unique effect. There are no items that cancel each other out.