Counter representation command
With circled numbers of package pifont only twenty (or ten) values for the footnote counter can be used. Thus there should be a warning or error, if the
footnote counter is out of this range.
For this purpose, we define a counter representation command \circnum that behaves similar to \alph:
\documentclass{article}
% smaller image for TeX.SX
\usepackage[paperwidth=20mm,paperheight=20mm,margin=1mm]{geometry}
\usepackage{pifont}
\makeatletter
\newcommand*{\circnum}[1]{%
\expandafter\@circnum\csname c@#1\endcsname
}
\newcommand*{\@circnum}[1]{%
\ifnum#1<1 %
\@ctrerr
\else
\ifnum#1>20 %
\@ctrerr
\else
\ding{\the\numexpr 171+(#1)\relax}%
\fi
\fi
}
\makeatother
\renewcommand*{\thefootnote}{\circnum{footnote}}
\begin{document}
\footnote{First}
\dots
\addtocounter{footnote}{18}
\footnote{Last}
% \footnote{Error}
\end{document}

If the latest footnote is activated, it throws an error:
! LaTeX Error: Counter too large.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.30 \footnote{
Error}
?
Reset counter for each page
If the footnote counter is not reset from time to time, it will likely run out of its allowed range. Then it might be helpful to reset the footnote counter for each page. There are several packages for this purpose:
Package footmisc with option perpage:
\usepackage[perpage]{footmisc}
Package perpage:
\usepackage{perpage}
\MakePerPage[1]{footnote}
Downside: The footnote is reset at the second LaTeX run, causing errors in the first run.
Package zref-perpage:
\usepackage{zref-perpage}
\zmakeperpage{footnote}
The following example file uses the latest method:
\documentclass{article}
% smaller image for TeX.SX
\usepackage[paperwidth=20mm,paperheight=20mm,margin=1mm]{geometry}
\usepackage{zref-perpage}
\zmakeperpage{footnote}
\usepackage{pifont}
\makeatletter
\newcommand*{\circnum}[1]{%
\expandafter\@circnum\csname c@#1\endcsname
}
\newcommand*{\@circnum}[1]{%
\ifnum#1<1 %
\@ctrerr
\else
\ifnum#1>20 %
\@ctrerr
\else
\ding{\the\numexpr 171+(#1)\relax}%
\fi
\fi
}
\makeatother
\renewcommand*{\thefootnote}{\circnum{footnote}}
\begin{document}
\footnote{First}
\dots
\addtocounter{footnote}{18}
\footnote{Last}
% \footnote{Error}
\newpage
\footnote{New page}
\end{document}
The second page:

\@circnummore expandable to get it work insidelongtable. – Heiko Oberdiek Jul 08 '18 at 13:58