Error: Counter too large! with \alph{cntr} and cntr>26

The alphalph package by H. Oberdiek provides the second numbering system out of the box. As usual with Heiko's packages, a counter representation such as

\renewcommand{\thesection}{\alphalph{\value{section}}}

is completely expandable.

And, as you remark in comments, with

\makeatletter
\newalphalph{\alphmult}[mult]{\@alph}{26}
\makeatother
\renewcommand{\thesection}{\alphmult{\value{section}}}

you get your preferred way. Of course, replace section with the counter you need.

Should you happen to use biblatex for your bibliography, you may use its \mknumalph macro for counter values up to 702 ("zz").

\documentclass{article}

\usepackage{biblatex}

% Everything from here to \begin{document} only serves to avoid biber errors
\usepackage{filecontents}

\begin{filecontents}{\jobname.bib}
@misc{A01,
  author = {Author, A.},
  year = {2001},
  title = {Alpha},
}
\end{filecontents}

\addbibresource{\jobname.bib}

\nocite{*}

\begin{document}

\begin{enumerate}
\def\theenumi{\mknumalph{\value{enumi}}}
\setcounter{enumi}{24}
\item twenty-five
\item twenty-six
\item twenty-seven
\end{enumerate}

\end{document}