Here is a sketch of how you could do this.
Start with a concave function f. Consider now the interval $(1,1.1)$. We will try to modify f by increasing its value on this interval while preserving the values at the endpoints. Let's set aside smoothness for now.
What properties should the modified f have? Subadditivity is threateneded only in cases where the $x + y$ happens to land inside $(1,1.1)$. Because of the concavity up to 1, we can see that it suffices to ensure that the new variant $f_1$ of f satisfy:
$$f_1(1 + \epsilon) \le f(1) + f(\epsilon)$$
Basically, we can move the subadditivity condition to the boundary because of the known concavity property of f.
Thus, $f_1$ can be chosen arbitrarily subject to taking the correct boundary values and to being bounded from above by the expression indicated. Assuming strict concavity, there is some free space between the actual current function $f$ and the upper bound for $f_1$ given by the equation. We can therefore choose a $f_1$ that works. Maintaining the condition of being increasing is not problematic -- even if it were, we could just add a huge coefficient linear function to $f_1$ and make it increasing. Smoothness is the relatively harder part, but it could be fixed using bump functions.
EDIT: There's another constraint, that again is achievable: subadditivity when both $x$ and $x + y$ are in the interval $(1,1.1)$.
I also think that functions such as $2x + \sqrt{x}e^{-(x-1)^2}$ or variants thereof may work directly, but I don't know of any easy analytical proof of it.