How can I reference the current (sub)section number?

You mean something like this? We simply use the fact that the "lower" section number is always 0 when that sectioning level is not used at the moment.

documentclass{report}

makeatletter
newcommand{getCurrentSectionNumber}{%
  ifnumc@section=0 %
  thechapter
  else
  ifnumc@subsection=0 %
  thesection
  else
  ifnumc@subsubsection=0 %
  thesubsection
  else
  thesubsubsection
  fi
  fi
  fi
}

makeatother

begin{document}

chapter{My Example}
A: Current number is getCurrentSectionNumber %1

section{Section One-point-One}
    B: Current number is getCurrentSectionNumber % 1.1
    subsection{subsection one-point-one-point-one}
        C: Current number is getCurrentSectionNumber % 1.1.1
    subsection{subsection one-point-one-point-two}
        D: Current number is getCurrentSectionNumber % 1.1.2
section{Section Two}
    E: Current number is getCurrentSectionNumber % 1.2
    subsection{subsection one-point-two-point-one}
        F: Current number is getCurrentSectionNumber % 1.2.1
    subsection{subsection one-point-two-point-two}
        G: Current number is getCurrentSectionNumber % 1.2.2

end{document}

Here is another way compared to the good answer from tohecz, using simply the fact that the label command refers to the last sectioning command called.

documentclass{book}

usepackage[hidelinks]{hyperref}
newcounter{GN}
setcounter{GN}{0}
newcommand{GetNumber}[0]{stepcounter{GN}label{GN:theGN}ref{GN:theGN}}

begin{document}

chapter{Foo}
Current number is: GetNumber
section{Bar}
Current number is: GetNumber
subsection{Baz}
Current number is: GetNumber
subsection{Baz2}
Current number is: GetNumber
section{Bar2}
Current number is: GetNumber

end{document}

And here the output:

EDIT Here is an improved version of my MWE to show the robustness of the method to the equation and float.

documentclass{book}

usepackage{float}
usepackage[hidelinks]{hyperref}
newcounter{GN}
setcounter{GN}{0}
newcommand{GetNumber}[0]{stepcounter{GN}label{GN:theGN}ref{GN:theGN}}

begin{document}

chapter{Foo}
Current number is: GetNumber
section{Bar}
Current number is: GetNumber
subsection{Baz}

begin{equation}c=a+bend{equation}

noindent
Current number is: GetNumber
subsection{Baz2}

begin{figure}[H]
caption{Figure to test the robustness}
end{figure}

Current number is: GetNumber
section{Bar2}
Current number is: GetNumber

end{document}

and here is its output, however as pointed out by tohecz and egreg in the comments, this solution works only if you call GetNumber out of an environment where a counter has been refstepped.

You have to examine each counter from top down and stop when you reach one that has the value zero.

documentclass{book}
usepackage{xparse}
ExplSyntaxOn
NewDocumentCommand{getCurrentSectionNumber}{}
 {
  jvri_get_current:
 }
% the list of examined counters
clist_const:Nn c_jvri_counters_clist 
 {
  chapter,section,subsection,subsubsection,paragraph,subparagraph
 }

% for storing the previous counter name
tl_new:N l_jvri_previous_tl

cs_new_protected:Npn jvri_get_current:
 {
  tl_set:Nn l_jvri_previous_tl { part } % for safety
  clist_map_inline:Nn c_jvri_counters_clist
   {
    int_compare:nTF { arabic{##1} = 0 }
     {% if the counter is zero, break and use the former the<counter>
      clist_map_break:n { use:c { the l_jvri_previous_tl } }
     }
     {% store the counter's name
      tl_set:Nn l_jvri_previous_tl { ##1 }
     }
   }
 }
ExplSyntaxOff

begin{document}

Zero: getCurrentSectionNumber (will have no output)

chapter{My Example}
A: Current number is getCurrentSectionNumber

section{Section One-point-One}
B: Current number is getCurrentSectionNumber

subsection{subsection one-point-one-point-one}
C: Current number is getCurrentSectionNumber

subsection{subsection one-point-one-point-two}
D: Current number is getCurrentSectionNumber

section{Section Two}
E: Current number is getCurrentSectionNumber

subsection{subsection one-point-two-point-one}
F: Current number is getCurrentSectionNumber

subsection{subsection one-point-two-point-two}
G: Current number is getCurrentSectionNumber

chapter{Again}
H: Current number is getCurrentSectionNumber

end{document}

This is easily extendable to manage also lower levels.

You'll see that after “Zero:” there will be nothing; it can be added a check whether the final sequence is empty.

If you are able to retrieve the current sectioning level (lets call it currsecnumdepth), you could also use plain old ifcase:

newcommand*{getCurrentSectionNumber}{ifcasecurrsecnumdepththechapterorthesection
orthesubsectionorthesubsubsectionortheparagraphorthesubparagraphfi}

In the following example, I provided a (admittedly unpleasent) way to make sure that the premise of my above sentence holds:

documentclass{report}

makeatletter
newcountcurrsecnumdepth
%verbose from the standard classes, except line 4
renewcommand{chapter}{if@openrightcleardoublepageelseclearpagefi
                        thispagestyle{plain}%
                        global@topnumz@
                        globalcurrsecnumdepthz@
                        @afterindentfalse
                        secdef@chapter@schapter}
%verbose from latex.ltx, except line 2
renewcommand{@startsection}[6]{%
  globalcurrsecnumdepth=#2
  if@noskipsec leavevmode fi
  par
  @tempskipa #4relax
  @afterindenttrue
  ifdim @tempskipa <z@
    @tempskipa -@tempskipa @afterindentfalse
  fi
  if@nobreak
    everypar{}%
  else
    addpenalty@secpenaltyaddvspace@tempskipa
  fi
  @ifstar
    {@ssect{#3}{#4}{#5}{#6}}%
    {@dblarg{@sect{#1}{#2}{#3}{#4}{#5}{#6}}}}
makeatother

newcommand*{getCurrentSectionNumber}{ifcasecurrsecnumdepththechapterorthesection
orthesubsectionorthesubsubsectionortheparagraphorthesubparagraphfi}

begin{document}

chapter{My Example}
A: Current number is getCurrentSectionNumber

section{Section One-point-One}
    B: Current number is getCurrentSectionNumber
    subsection{subsection one-point-one-point-one}
        C: Current number is getCurrentSectionNumber
    subsection{subsection one-point-one-point-two}
        D: Current number is getCurrentSectionNumber
section{Section Two}
    E: Current number is getCurrentSectionNumber
    subsection{subsection one-point-two-point-one}
        F: Current number is getCurrentSectionNumber
    subsection{subsection one-point-two-point-two}
        G: Current number is getCurrentSectionNumber

end{document}