Solution using titletoc
I find that the titletoc package provides the best options for customization — as long as you really know what you want! Here is an implementation for the chapter entry:
\documentclass{memoir}
\usepackage{titletoc}
% This command "floats" the page number to the far right of the current line.
\newcommand\floatcontentspage{%
\hbox to 0pt{%
\hspace{-\leftskip}\hspace{\textwidth}\hspace{-\rightskip}%
\textbf{\contentspage}}%
}
\titlecontents
{chapter} % 1. <section>
[0pt] % 2. <left> indent (none here)
{} % 3. <above code>: i.e. vertical space
{% % 4. <numbered-entry-format> for numbered chapters
\contentspush{\floatcontentspage%
\textbf{\chaptername~\thecontentslabel}%
\hspace{1.5em}}}
{} % 5. <numberledd-entry-format> for unnumbered chapters
{} % 6. <filler-page-format> no filler needed
[\vspace{\baselineskip}] % 7. <after code>: i.e. vertical space
\begin{document}
% I am not exactly sure what I should do here, but \chapternumberline has
% \@chapapp@head which is undefined when I use titletoc... This is an
% unexplained hack to get it working
\renewcommand\chapternumberline[1]{\numberline{#1}}
\tableofcontents*
\chapter{First chapter}
\chapter{Second chapter, in which many wonderful things happen, and then other things.}
\end{document}

Explanation:
I first define the command \floatcontentspage:
\newcommand\floatcontentspage{%
\hbox to 0pt{%
\hspace{-\leftskip}\hspace{\textwidth}\hspace{-\rightskip}%
\textbf{\contentspage}}%
}
This is intended to be inserted at the start of a contents line after the left indent (of \leftskip). It typesets its contents in a box of width 0pt so that it does not affect the rest of the line. I first insert some space to total \textwidth - \leftskip - \rightskip to obey the current margins, then set the page number using the \contentspage command which I have wrapped in \textbf{} to match the style of the original poster. This will push the page number out to the right after the inserted space.
I then use the \titlecontents command to define the formatting of the contents line. It takes arguments in the following order:
{chapter} specifies the section type.
[0pt] is the optional left indent (none here: you would probably want this for section etc.).
{} There is no above code (vertical space).
- This is the main entry for numbered chapters discussed in a moment.
{} I have not defined anything for unnumbered chapters.
{} There is no filler-page-format since we have already typeset the pagenumber.
[{\vspace{\baselineskip}] I have added some space after the chapter entry. (You might like to do this before instead.)
The work is done by argument 4.:
{% % 4. <numbered-entry-format> for numbered chapters
\contentspush{\floatcontentspage%
\textbf{\chaptername~\thecontentslabel}%
\hspace{1.5em}}}
This calls \contentspush which inserts some text and then indents the remaining lines accordingly. We insert our \floatcontentspage command first which will float the page number to the right but take no space, then we insert the \chaptername and \thecontentlabel (in \textbf{}) which gives "Chapter 1" etc. Finally we insert some space to separate the title from the number.
Finally, I had to redefine \chapternumberline to call \numberline: I do not understand exactly why, but \chapternumberline uses \@chapapp@head which is undefined...
As you can see, the titletoc package allows for great flexibility, but requires that you know exactly what you want, and you generally must define all of your entries (I have only provided the chapter definition here. How do you want the sections defined?) Provided that you have a clear idea, though, it is the best tool in my opinion.
\chaptercommand. – Alan Munn Mar 02 '11 at 19:29