Enumerate List with Different Base number

\documentclass{article}

\usepackage{tikz}

\renewcommand*{\labelenumi}{\pgfmathparse{int(2^(\theenumi-1))} \pgfmathresult}


\begin{document}
    \begin{enumerate}
        \item One
        \item Two
        \item Three
        \item more
        \item more
        \item more
        \item more
        \item more
        \item more
        \item more
        \item more
        \item more
        \item more
    \end{enumerate}
\end{document}

Here's a LuaLaTeX-based solution, which works with the enumitem package. It defines an enumerated environment called powertwoenum, in which consecutive items are numbered as 1, 2, 4, 8, 32, etc. Items in a powertwoenum list may be cross-referenced via the usual \label-\ref mechanism.

% !TEX TS-program = lualatex
\documentclass{article}
\newcommand\powertwo{\directlua{tex.sprint(2^(\arabic{powertwoenumi}-1))}}
\usepackage{enumitem}
\newlist{powertwoenum}{enumerate}{1}
\setlist[powertwoenum]{label=\protect\powertwo.,ref=\protect\powertwo}


\begin{document}
\begin{powertwoenum}
  \item First
  \item Second
  \item Third
  \item Fourth
  \item Fifth \label{item:five}
\end{powertwoenum}

A cross-reference to item \ref{item:five}.

Resume the enumerated list.

\begin{powertwoenum}[resume]
  \item First
  \item Second
  \item Third
  \item Fourth
  \item Fifth
\end{powertwoenum}
\end{document}

A variant of Christian Hupfer's solution, but fully expandable:

\documentclass{article}

\usepackage{enumitem}
\usepackage{xparse}

\ExplSyntaxOn
\DeclareExpandableDocumentCommand{\basetwoenum}{m}
 {
  \basetwoenum_main:n { #1 }
 }
\cs_new:Nn \basetwoenum_main:n
 {
  \exp_args:Nc \basetwoenum_eval:n { c@#1 }
 }
\cs_new:Nn \basetwoenum_eval:n
 {
  \fp_eval:n { 2 ** (#1-1) }
 }
%% for keeping enumitem happy
\cs_set_eq:cN { @basetwoenum } \basetwoenum_main:n
\ExplSyntaxOff

\makeatletter
\AddEnumerateCounter{\basetwoenum}{\@basetwoenum}{000}
\makeatother 

\begin{document}

\begin{enumerate}[label={\basetwoenum*}]
  \item First
  \item Second
  \item Third
  \item Fourth
  \item Fifth
\end{enumerate}

Resume it

\begin{enumerate}[label={\basetwoenum*},resume]
  \item First
  \item Second
  \item Third
  \item Fourth
  \item Fifth
\end{enumerate}

\end{document}

This gives correct results up to 2⁵³; of course, if you plan to use more than ten items you have to accordingly increase the number of zeros in the final argument to \AddEnumerateCounter.