\sum caused \sqrt to not be slanted

If you look carefully you can see that the angle changes if the part below the root sign gets larger until it is vertical. Imho all math fonts will do it. So if you want a slant: make the content smaller, e.g. by using \nolimits:

\documentclass{article}

\begin{document}

\[
\sqrt{1+ \sqrt{1+ \sqrt{1 + \sqrt{1+ \sqrt{1+ \sqrt{1+x}}}}}} 
\]

\[
\sqrt{\sum\nolimits_{i = 1}^{n}{x^i}}
\]

\end{document}

The only math font package I know that has slanted surds at large sizes is mtpro2:

\documentclass{article}
\usepackage[lite]{mtpro2}

\begin{document}

\[
\SQRT{1+ \SQRT{1+ \SQRT{1 + \SQRT{1+ \SQRT{1+ \SQRT{1+x}}}}}}
\]

\[
\SQRT{\sum_{i = 1}^{n}{x^i}}
\]

\end{document}

Note that \SQRT should be used for this to work. Unfortunately, adapting this to other font families would be very complicated. The package changes all fonts to be Times like.

With MinionMath-Regular:

\documentclass{article}
\usepackage{unicode-math}
\setmathfont{MinionMath-Regular.otf}
\begin{document}

\[
  \sqrt{{x^i}}
\]
\[
  \sqrt{\sum_{i = 1}{x^i}}
\]
\[
  \sqrt{\sum^{n}{x^i}}
\]
\[
  \sqrt{\sum_{i = 1}^{n}{x^i}}
\]

\end{document}