There's no real definitive answer to this question but here are some of my thoughts on the issue. Many people when they propose adding new amino acids look at polar amino acids (such as the ethers, hyroxyproline or hydroxylysine proposed in this thread). These polar amino acids seem much more interesting than the simple aliphatic amino acids because they can do so much more stuff (like hydrogen bond, bind metals, aid in catalysis, etc). However, the "boring" aliphatic residues are essential for a process much more important than catalysis, namely protein folding. In addition to being a prerequisite for any functional role the protein plays (including catalysis), proper protein folding is also important because misfolded proteins are toxic to the cell. Thus, evolution has very strongly selected for factors that promote protein folding.
Protein folding is largely driven by the hydrophobic effect, and the protein will fold to conceal its nonpolar residues within the interior of the protein and expose its polar residues on the exterior. Therefore, the packing of non-polar residues within the interior of the protein is highly important to its stability. In addition, evolution tunes the stability of proteins so that instead of being completely rigid, they undergo important conformational changes that contribute to their functions, like catalysis or binding.
A similar principle also applies to many protein-protein interactions where the association of two complementary hydrophobic surfaces provides most of energy to stabilize the binding interaction. In most cases, the hydrophobic effects drives the stability of such interactions while hydrogen bonding and salt bridge formation provide the specificity for such interactions. (Getting away from the protein world, the association of two strands of DNA into a double helix illustrates this point well. The hydrogen bonding between the bases do not contribute much to the thermodynamic stability of the double helix; there is very little energetic difference between the hydrogen bonding of the bases to water in ssDNA and the hydrogen bonding between complementary bases in dsDNA. Rather, the stacking between the aromatic rings of the bases, sequestering these fairly hydrophobic parts within the interior of the double helix, that provides the energy to drive the formation of dsDNA. The base pairing merely insures specificity for the interactions.)
Therefore, nature requires a number of nonpolar amino acids of varying shapes and sizes in order to be able to create well-packed protein interiors and protein-protein interaction interfaces. The variety of hydrophobic amino acids is also important for tuning the strength of these folding and binding interactions. Of course, making a variety of hydrophobic amino acids is difficult because synthesizing aliphatic chains of different lengths requires carbon-carbon bond formation. Thus, perhaps having the closely related set of isoleucine, leucine, and valine (in addition to the aromatics and other nonpolar residues) represent some sort of compromise between systems that lack variety in the shapes of the hydrophobic residues versus systems that require very complex biosynthetic schemes.