Here's a way of making the \fakefigure macro suggested in the question work. Well, at least it survives minimal testing in combination with Werner's answer on absolute placement.
The proposed redefinition of \caption won't work because capt-of defines \captionof in terms of \caption. However, it is possible to first save the definition of \caption and then use the saved definition when redefining \caption.
\global\let\oldcaption\caption
\renewcommand\caption{\def\@captype{figure}\oldcaption}% modified from capt-of
I've only allowed for figure type elements here. If you need other kinds of floats to be converted to non-floats, you'd want to reinstitute the optional argument provided by capt-of.
We can then combine this with Mico's suggestion to create the \fakefigure{} macro:
\makeatletter
\global\let\oldcaption\caption
\newcommand{\fakefigure}[1]{%
\begin{minipage}{\textwidth}
\renewcommand\caption{\def\@captype{figure}\oldcaption}% modified from capt-of
#1
\end{minipage}%
}
\makeatother
This then allows us to write something like
\placetextbox{.35}{,3}{%
\fakefigure{%
\includegraphics[height=.2\textheight]{tiger}%
\caption{abc}%
\label{fig:abc}%
}%
}
to place TeX's standard tiger at an arbitrary position on the page.

\documentclass{article}
\usepackage{graphicx,kantlipsum,eso-pic}
\makeatletter
\global\let\oldcaption\caption
\newcommand{\fakefigure}[1]{%
\begin{minipage}{\textwidth}
\renewcommand\caption{\def\@captype{figure}\oldcaption}% modified from capt-of
#1
\end{minipage}%
}
\makeatother
% from Werner's answer at https://tex.stackexchange.com/a/24675/
\newcommand{\placetextbox}[3]{% \placetextbox{<horizontal pos>}{<vertical pos>}{<stuff>}
\setbox0=\hbox{#3}% box
\AddToShipoutPictureFG*{% Add <stuff> to current page foreground
\put(\LenToUnit{#1\paperwidth},\LenToUnit{#2\paperheight}){\vtop{{\null}\makebox[0pt][c]{#3}}}%
}%
}
\begin{document}
\placetextbox{0.5}{0.5}{\fbox{\Huge\textsf{This is my text.}}}
\placetextbox{0.5}{1}{\Huge\texttt{Here is another piece of text.}}
\placetextbox{0.1}{0.1}{\Large $\mathcal{A}_1$}
\placetextbox{.35}{,3}{%
\fakefigure{%
\includegraphics[height=.2\textheight]{tiger}%
\caption{abc}%
\label{fig:abc}%
}%
}
\kant[1-3]
\end{document}
\captionof{figure}{caption text}. – Bernard May 20 '16 at 21:50\def\caption{\captionof{fig:abc}and\renewcommand\label[1]{}would it work? – k_g May 20 '16 at 21:53figureenvironment. You need not to use it. You don't even have to have a caption if you don't want. If you do want, that's a secondary issue. The main point is: if you don't want something to float, don't make it a float. A figure is a float. Floats, by design, float. That's kind of their raison d'être. – cfr May 20 '16 at 22:49figureenvironments". No way, no how. Time to start looking into using the\captionofmacro. – Mico May 20 '16 at 23:37as few changes as possibleincludes making it not a float. – cfr May 20 '16 at 23:48\fakefigurethen you are talking about not usingfigure, right? So that involves conversion from float to non-float. – cfr May 20 '16 at 23:51\renewcommand\caption ...like that because\captionofis defined in terms of\captionby thecapt-ofpackage. – cfr May 21 '16 at 00:03