Is there an always maximal non-constant function?

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I wonder, if there is an function (quite obviously (?) no $\mathbb R$-$\mathbb R$- function) which is non-constant (actually it should be an increasing function) and which is maximal everywhere?

Nice to have (or a necessary condition for the existence for such a function?): The maximum-property may be dependent from an "observer". I am not sure, if this additional condition makes the problem more complex or if this condition may the problem solveable in the first place.