Is there any hypothesis on the minimum 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: