  Contextuality is a fundamental feature of quantum theory and a necessary
resource for quantum computation and communication. It is therefore important
to investigate how large contextuality can be in quantum theory. Linear
contextuality witnesses can be expressed as a sum $S$ of $n$ probabilities, and
the independence number $\alpha$ and the Tsirelson-like number $\vartheta$ of
the corresponding exclusivity graph are, respectively, the maximum of $S$ for
noncontextual theories and for the theory under consideration. A theory allows
for absolute maximal contextuality if it has scenarios in which
$\vartheta/\alpha$ approaches $n$. Here we show that quantum theory allows for
absolute maximal contextuality despite what is suggested by the examination of
the quantum violations of Bell and noncontextuality inequalities considered in
the past. Our proof is not constructive and does not single out explicit
scenarios. Nevertheless, we identify scenarios in which quantum theory allows
for almost absolute maximal contextuality.
