Why is 1-methylhexane not a legitimate name for a compound?

It is not a legitimate chemical name because the methyl group on carbon 1 increases the length of the carbon chain to seven carbon atoms thus making the parent compound heptane

Explanation:

Chemical compounds are named according to rules in the IUPAC nomenclature.

In naming organic compounds, parts of rules is to find out the parent or root compound. This is determined by finding out the longest carbon chain and then checking if it contains single or multiple bonds.  (that is if it is an alkane- with single bonds, alkene-with double bonds or an alkyne- with triple bonds)

From the question the parent or root name is an alkane containing 6 (Hex-) carbon atoms, while the methyl group is  substituent

(CH₃)-CH₂-CH₂-CH₂-CH₂-CH-CH₃

However the alkyl substituent on carbon 1 increases the the length of the carbon chain to 7 (Hept-) instead of 6. Hence the correct IUPAC nomenclature is heptane and not 1-methylhexane.