Let $c_i$ be arbitrary positive constants. Is there a way to upper bound $\text{Var}(\max_i \{X_i + c_i\})$ by some constant multiple of $\text{Var}(\max_i \{X_i \})$ that is independent of the values of $c_i$'s?
My gut feeling is that this is possible because the $c_i$'s are constants and they do not affect the randomness seen by the variance.
(In my application, $X_i = \sum_{k=1}^i Y_k$ where $Y_i$ are subexponential iid random variables.)
Let $X_m = \sum_{k=1}^m Y_k$ where $Y_k$ are subexponential i.i.d random variables of negative mean $-1$. Then $$\text{Var}(\max_{m \in [1,n]} \{X_m \})=O(1)$$ (This is easiest to see if $Y_k$ are uniform on $[-2,0]$, say). However, if we take $c_m=m$, then $$\text{Var}(\max_{m \in [1,n]} \{X_m + c_m\})$$ is of order $n$.