I suppose the title virtually answers the question. What components in the atmosphere can result in a chemical reaction?
The degree to which these reactions are a concern can motivate one to exclude them, hence inert atmosphere. Here are some examples. Solutions of NaOH lose their titer, ethers form peroxides, benzaldehyde forms benzoic acid, and acetyl chloride forms HCl and acetic acid if exposed to air. What causes these reactions? Those are the most common reactants in air. What are the reactions? I would expect that using an inert atmosphere is to exclude one or all of them.