A partial ordered set $(P,\preceq)$ (i.e. a set $P$ endowed with a operation $\preceq$ which is reflexive, antisymmetric and transitive) is said to be a join-semilattice whenever for all $x,y \in P$ there exists their least upper bound $x \vee y \in P$.
Then, given a join-semilattice $(P,\preceq)$, we fix a subadditive function $f\colon P\to \mathbf{R}$, meaning that $$ \forall x,y \in P,\,\,f(x\vee y)\le f(x)+f(y), $$ and such that $f(x)<f(y)$ for some $x\preceq y$. [<- Thank you Ramiro]
Question: Is it possible that $P$ is an infinite countable set and for all $x \in P$ and $\varepsilon >0$ there exist $a_1,\ldots,a_k \in P$ for which $x=\bigvee a_i$ and $f(a_1)<\varepsilon,\ldots,f(a_k)<\varepsilon$?
Yes, this is possible. For instance, let $P$ be the set of finite unions of intervals in $\mathbb{R}$ with rational endpoints, ordered by inclusion, and let $f:P\to\mathbb{R}$ be Lebesgue measure. Since any interval with rational endpoints can be partitioned into finitely many arbitrarily small intervals with rational endpoints, this satisfies your condition.