Use update limits=false
. Modifying the bounding box is also possible, but not what you want in this context -- you want to change the entire appearance of the axis (unless I am mistaken). A solution is
\documentclass{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
no markers,
trig format plots=rad,
restrict y to domain=-1:10,
]
\addplot {sin(x)};
\addplot+[update limits=false] {exp(x)};
\end{axis}
\end{tikzpicture}
\end{document}

Note that I had to add restrict y to domain
due to an internal weakness with the floating point arithmetics in pgfplots.
Alternatively, you can use an environment to disable updates to the axis limits. This may be more effective in order apply the modification to "all following plots":
\documentclass{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
no markers,
trig format plots=rad,
restrict y to domain=-1:10,
]
\addplot {sin(x)};
\begin{pgfplotsinterruptdatabb}
\addplot {exp(x)};
\end{pgfplotsinterruptdatabb}
\end{axis}
\end{tikzpicture}
\end{document}
Note that overlay
addresses a different use-case: it excludes graphical elements from the bounding box only, but it does not affect how axis limits are chosen. Thus, overlay
instead of update limits=false
has no effect since the axis limits would still be chosen as before, and since the axis lines define the bounding box, the overlay
feature has no effect.
However, overlay
has an effect if you hide the axis lines:
\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}
\begin{document}
This is the figure and its BB:
\fbox{%
\begin{tikzpicture}
\begin{axis}[
no markers,
trig format plots=rad,
restrict y to domain=-1:10,
hide axis,
]
\addplot {sin(x)};
\addplot+[overlay] {exp(x)};
\end{axis}
\end{tikzpicture}}
\end{document}

Note that both figures have been respected during the axis limit computation (which can be seen by the fact that the y axis is scaled down). But the second plot does not affect the bounding box as can be seen by the \fbox
command; it even crosses the text.
Thus: update limits=false
controls how a single plot affects the axis limits (which has implications for the scaling and the descriptions). The overlay
key is useful if it comes to the graphical elements and their bounding box.