\left[ and \right] determine the size of the delimiters by looking of the content between them, including the \overbrace. To have them ignore the brace, put it inside a \smash. This is demonstrated by the second line in the example below.
Since you mention that the brace should not effect the layout in any way, you might also want to ignore the width of the annotation added to the brace. You can achieve this by putting the annotation inside a box of zero width, \makebox[0pt]. This is shown by the third line.
The fourth line finally combines the two. Be aware that both \smash and \makebox can lead to undesirable results here, as they disable the automatic adjustment of spaces and positioning. With \smash, you can end up with too small brackets on the sides. With \makebox, the annotation text can easily overlay other math symbols in the equation.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align}
\left[\frac{a}{b}+\overbrace{\phi\times\rho}^{\text{my annotation}}+\int_0^\infty\right] \\[1em]
\left[\frac{a}{b}+\smash{\overbrace{\phi\times\rho}^{\text{my annotation}}}+\int_0^\infty\right] \\[1ex]
\left[\frac{a}{b}+\overbrace{\phi\times\rho}^{\text{\makebox[0pt]{my annotation}}}+\int_0^\infty\right] \\[1em]
\left[\frac{a}{b}+\smash{\overbrace{\phi\times\rho}^{\text{\makebox[0pt]{my annotation}}}}+\int_0^\infty\right]
\end{align}
\end{document}
$[a + \overbrace{b}^{c}]$. Then the brackets are there in normal size. – dexteritas Nov 30 '19 at 16:19