What is the smallest number of amino acids required for life?
You can divide the 22 (including selenocysteine and pyrrolysine)  proteinogenic amino acids into broad groups of similar amino acids. There are the hydrophobic amino acids like trypthophane, valine and leucine, the charged amino acids like glutamate and arginine and the polar amino acids like serine and threonine. There are some amino acids with unique features like cysteine which can form disulfide bonds.

Some amino acids are very similar, for example isoleucine and leucine, it is plausible that one of those would suffice to create most protein folds.

There are several examples of proteins designed with a smaller alphabet of amino acids, one example is the E. coli orotate phosphoribosyltransferase (Akanuma et al., 2002). The simplified enzyme consists of only 13 different amino acids and 88% of it are composed of only nine different amino acids. Even after those drastic changes the enzyme still folds correctly and has enzymatic activity.

There is one study (Fan and Wang, 2003) that tried to answer exactly the question you asked. They came to the conclusion that around 10 amino acids are necessary to create properly folding proteins:


  First, we study the minimum sequence complexity that can reserve the
  necessary structural information for detection of distantly related
  homologues. Second, we compare the ability of designing foldable model
  sequences over a wide range of reduced amino acid alphabets, which ﬁnd
  the minimum number of letters that have the similar design ability as
  20. Finally, we survey the lower bound of alphabet size of globular proteins in a non-redundant protein database. These different
  approaches give a remarkably consistent view, that the minimum number
  of letters required to fold a protein is around ten.




Akanuma, S., Kigawa, T. &amp; Yokoyama, S. Combinatorial mutagenesis to restrict amino acid usage in an enzyme to a reduced set. Proceedings of the National Academy of Sciences 99, 13549 -13553 (2002).

Fan, K. &amp; Wang, W. What is the minimum number of letters required to fold a protein? J. Mol. Biol. 328, 921-926 (2003).