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.
Flicking through Sets on BrickLink, sorted by number of parts (and ignoring small-scale items such as "Advent Calendar" models), I identified the 61-part set 4736-1 "Freeing Dobby" from 2010, which included no new parts - and, as such, no unique parts.
The rarest part of the set appears to be Tan Tile 1 x 2 with Groove with HP Sock Pattern, which appears in two other sets
This then proves that not all Lego sets include unique parts.
I really doubt that every single Lego set on the planet has at least one unique piece, and besides, I think that's up to you to decide which pieces are “unique” and which are not. Every single set that I try thinking of seems to have a unique piece, but I bet the real reason for that is because I have just never seen how many sets have that same piece, therefore I think it's rare when it's really not. One group of sets that is always guaranteed to have rare pieces is the Lego Minifigures series.
So the short answer is most likely no.
My understanding is that it costs Lego quite a bit to create a new mold, so it's in their interest to reuse their pieces a lot.
The question of uniqueness has already a lot of answers, I will concentrate on this question from the text
If I had 100 lego sets would I be able to fully build a lego set that I didn't have?
There is another factor, namely time, to consider. While the basic LEGO bricks remain unchanged over the decades, other items are retired and new ones are introduced. The style of doors, fences, and windows has changed considerably over time, and you need sets covering a certain period to rebuild another period set. 100 sets of the 1980ies will not help you in rebuilding a modern model, and 100 contemporary sets will not be sufficient to rebuild an original Black Seas Barracuda from 1989 because this brick was phased out in 2007 and in yellow, it was last seen in 2002.
Degrees of uniqueness
Unique molds
A special mold is the most expansive form of uniqueness that involves a lot of investment by the LEGO company. Really unique molds are rare, but they occur, e.g., in licenced sets, as minifigure parts or accessories. For a long time, the One Ring from one LoTR LEGO set was unique, until recently revied as a wedding ring in the Brickheads series.
Unique colourings
This kind of uniqueness is somewhat more frequent: You can have known element in a unique or rare colour for some model.
Unique prints
Even more frequent are unique prints on known elements of known colours, including the bodies, legs, and heads of minifigures, or special printings bricks.
Stickers
Stickers from sticker sets are really often unique. It is not unusual for them to contain a sticker with the official set number.
