Yes (to the question in your post, not in the title). Here is an example: Choose your favourite prime $p$, your favourite positive number $r\notin p^\mathbb{Q}$, and define the following absolute value on the polynomial ring $\mathbb{Q}[t]$:
$$|\sum_{i=0}^n a_i t^i| := \max (|a_i|_p r^i)$$
where $|\cdot|_p$ is the usual $p$-adic absolute value on $\mathbb{Q}$. Extend it multiplicatively to the field of fractions $K:=\mathbb{Q}(t)$. We have
$$|K| = \{0\} \cup p^\mathbb{Z}\cdot r^\mathbb{Z}.$$
In particular both $p$ and $r$ are in $|K^*|$, but if there were an $a\in \mathbb{R}$ with $p \in a^\mathbb{Q} \ni r$, it would follow that $r \in p^\mathbb{Q}$, contradiction.
Note that of course there are other non-archimedean (and non-trivial) absolute values on that field such that the value group $|K^*|$ is contained in $a^\mathbb{Q}$. A follow-up question would be: Are there any fields $K$ such that every non-trivial nonarchimedean value on $K$ has a value group that is not contained in some $a^\mathbb{Q}$?