Can a saturated hydrocarbon have side chains?
Standard organic chemists’ understanding is that saturated/unsaturated is a word pair entirely independent of branched/linear/cyclic. Thus, all six combinations are possible:


branched/saturated (e.g. neopentane or 2,2-dimethylpropane)
branched/unsaturated (e.g. 2,3-dimethylbut-2-ene)
linear/saturated (e.g. n-pentane)
linear/unsaturated (e.g. hex-3-ene)
cyclic/saturated (e.g. cyclohexane)
cyclic/unsaturated (e.g. cyclohexene)


I chose these examples for a reason. The central carbon atoms in both neopentane and 2,3-dimethylbut-2-ene are only bonded to other carbon atoms while all other carbons in the branched examples are methyl groups ($\ce{CH3}$). Yet, no organic chemist will classify neopentane as ‘unsaturated’ — so clearly there must be an additional criterion other than ‘maximise the number of bonds to hydrogen atoms’.

On the other hand, hex-3-ene has almost maxed out its number of hydrogen atoms: two carbons with three, two carbons with two and only two carbons which are lacking a hydrogen each. But that does not make it any less unsaturated. Note that cyclohexane and hex-3-ene have the same formula $\ce{C6H12}$ and thus the same carbon/hydrogen ratio. Yet, the cyclic compound is considered saturated and the linear isn’t.

Thus, forget about the maximum hydrogen definition. A saturated molecule is one that is made up of single bonds only.

Note: Of course, cyclic compounds can also be branched or ‘substituted’. But that doesn’t add too much.