IUPAC nomenclature of aspartame
Aspartame is (a methyl ester of) a dipeptide. Therefore, you could use the nomenclature of amino acids to find a descriptive name for aspartame.

However, since you are asking about the correct order of the individual groups, apparently, you want to generate the fully systematic name according to nomenclature for organic chemistry.

Since there are various groups, you have to identify the principal characteristic group in accordance with the seniority of classes and suffixes first. In this case, the characteristic group with the highest seniority is the carboxylic acid group.

The principal characteristic group is chosen for citation at the end of a name by means of a suffix or a class name (or implied by a trivial name). Since there is only one carboxylic acid group in aspartame, the principal characteristic group also decides the choice of a principal chain. Thus, the parent structure of aspartame is butanoic acid.

The remaining substituents are expressed as prefixes in the name in alphabetical order. In this case, the remaining substituents are one amino group, one oxo group, and the complex substituent group [(2⁠S)-1-methoxy-1-oxo-3-phenylpropan-2-yl]amino.

Therefore, the complete systematic name of aspartame is (3⁠S)-3-amino-4-{[(2⁠S)-1-methoxy-1-oxo-3-phenylpropan-2-yl]amino}-4-oxobutanoic acid.