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Here's a made up example. Let's say that I have two factors, energy drink (Gatorade vs water) and gender (male and female). The outcome variable is mile time (continuous). A significant two-way interaction between energy drink and gender would suggest that the effect of energy drink on mile time differs between males and females. I have no problem understanding this conceptually.

However, let's say that I add a third factor, age (40+ vs 0-40). I understand that a significant three way interaction would tell you that the interaction of energy drink and gender differs across the levels of age. However, I'm not sure what the utility of interpreting the three way interaction term is. What is it telling me conceptually/ What's the point?

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Your techinical interpretation is quite correct.

So, let's say that Gatorade was associated with faster mile times than water, and this association was larger in males than in females.

The three-way interaction with age group may then tell you, for example, that this association disappeared entirely in the older age group, but was still evident in the younger age group. Or may be more pronounced in the older age group, or indeed that it was the same in both age groups.

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