How to typeset enumerations with asterisk?

Question

Minimal example:

\documentclass{article}

\begin{document}

\begin{enumerate}
  \item See below
  \item [2\textsuperscript{*}.] This isn't right. Or?
  \item [3*.] This neither.
  \item [4.*] Even worse.
  \item See above
\end{enumerate}

\end{document}

Shouldn't instead both the dot and the asterisk be directly at the number?

(I found this answer https://tex.stackexchange.com/a/298893/36836 but I am not familiar with the \up command. Is it just equivalent with \textsuperscript?)

Edit

I agree that the the title of a question marked as duplicate (Add asterisk after labels in enumerate) can be read to ask the same question. However my question added in the body of the question points to a more particular request that seems neither in the other question nor in any of the answer given to that question. I could change the title of my question slightly but that would render some of the answers given to my question look strange. So I leave it with this edit.

Edit 2

Based on a comment (How to typeset enumerations with asterisk?) I prefer as a solution:

\newcommand{\aster}{\makebox[0pt][l]{*}}

Answer 1

From your information I gather that you are trying to have some items stand out using stars, and you want it typeset directly above the dot. There are numerous ways to do this. Here I made a function, which do the counting for you. You could improve this using the enumitem package, so you don't need to write [\printnum] for every item. That all depends on how often you will need to do a list like this in your document.

\documentclass{article}
\usepackage{xparse}
\newcounter{itemC}
\NewDocumentCommand{\printNum}{s}{%
    \stepcounter{itemC}
    \IfBooleanTF{#1}%
    {%
        %Do Starred%
        \theitemC.\makebox[0pt]{*}%
    }{%
        %Do Unstarred%
        \theitemC.%
    }%
    }

\begin{document}

\begin{enumerate}
  \item [\printNum]  See below
  \item [\printNum*] This isn't right. Or?
  \item [\printNum*] This neither.
  \item [\printNum*] Even worse.
  \item [\printNum] See above
\end{enumerate}

\end{document}

Answer 2

I would personally never use any of 1.* or 1*. or 1{* above .}, all of them just look wrong. I would stick to the 1) and 1*) style. Also, I think you shouldn't have to number the list manually, so I coded a \staritem for you:

\documentclass{article}

\usepackage{enumitem}

\setlist[enumerate,1]{label=\arabic*)}

\makeatletter
\newcommand\staritem{\@noitemargtrue\@item[\arabic{enumi}*)]}
\makeatother

\begin{document}

Some text some text some text some text some text
some text some text some text some text some text.
\begin{enumerate}

\item Foo foo foo foo foo foo foo foo foo foo foo
 foo foo foo foo foo foo foo foo foo foo foo foo.

\staritem Bar bar bar bar bar bar bar bar bar bar
 bar bar bar bar bar bar bar bar bar bar bar bar.

\item Baz baz baz baz baz baz baz baz baz baz baz
 baz baz baz baz baz baz baz baz baz baz baz baz.

\end{enumerate}
Some text some text some text some text some text
some text some text some text some text some text.

\end{document}

Answer 3

Yet not sure to have fully understood, but loading the enumitem package, this gives an automatic label:

\documentclass{article}
\usepackage{enumitem, makebox}
\setlist[enumerate, 1]{label =\arabic*\rlap{\textsuperscript*}\protect\makebox*{\textsuperscript*}{.}}

\begin{document}

\begin{enumerate}%
\item Is this OK? 1
\end{enumerate}

\end{document} 

Answer 4

Assuming you want the asterisk over the top of the dot, with no spacing:

\documentclass{article}
\usepackage{calc}

\makeatletter
\newcommand{\aster}{%
    \newlength{\l@aster}%
    \settowidth{\l@aster}{*}%
    *\hspace{-\l@aster}%
}
\makeatother

\begin{document}

\begin{enumerate}
  \item See below.
  \item [2\textsuperscript{*}.] This isn't right. Or?
  \item [3*.] This neither.
  \item [4.*] Even worse.
  \item [5\aster{}.] How about this?


\end{enumerate}

\end{document}

I've called the new asterisk command \aster because I couldn't immediately think of anywhere else that's used. I suggest you pick your own name, pairing the new length \l@aster with it.

Personally I'd go for a different version (\item[*6.]):