  For an integer $n$, write $X_n(K)$ for the 4-manifold obtained by attaching a
2-handle to the 4-ball along the knot $K\subset S^3$ with framing $n$. It is
known that if $n< \overline{\text{tb}}(K)$, then $X_n(K)$ admits the structure
of a Stein domain, and moreover the adjunction inequality implies there is an
upper bound on the value of $n$ such that $X_n(K)$ is Stein. We provide
examples of knots $K$ and integers $n\geq \overline{\text{tb}}(K)$ for which
$X_n(K)$ is Stein, answering an open question in the field. In fact, our family
of examples shows that the largest framing such that the manifold $X_n(K)$
admits a Stein structure can be arbitrarily larger than
$\overline{\text{tb}}(K)$. We also provide an upper bound on the Stein framings
for $K$ that is typically stronger than that coming from the adjunction
inequality.
