Spiral approaching a circle

We choose a function for the radius that approaches 1. For example, we could use 1+2*exp(-0.1*\t):

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
    \draw [red] (0,0) circle [radius=1];
    \draw [domain=0:50,variable=\t,smooth,samples=500]
        plot ({\t r}: {1+2*exp(-0.1*\t)});
\end{tikzpicture}
\end{document}