Does each LEGO set include a unique piece? If I had 100 lego sets would I be able to fully build a lego set that I didn't have? Or would it always be impossible due to each set having at least one unique piece?

To further clarify, let's say I own every LEGO set except for one. Does the set I don't own always contain at least one brick/piece that can't be found in all the sets I do own? Of course color doesn't matter.
Most of sets doesn't contain unique elements, however there are exceptions. This (uniqueness) is especially common with minifigures or limited edition/availability sets.

The easiest set (polybag) to confirm the fact that only some sets have unique parts is to take 624210, which consists of 6 pcs. of iconic 2 x 4 bricks in Red, which were available in numerous sets. Another option, of a boxed set, is to take any set consisting of basic bricks, like 5576. Parts included in 5576 are widely available and can be found in multiple sets. 

However if you take something exclusive, like 41999. This set contains some unique parts available in this set only.