Weird spacing in aligned environment

I'd use an array environment, which is highly customizable.

\documentclass{article}
\usepackage{array} % for "\newcolumntype" directive
\newcolumntype{C}{>{{}}c<{{}}}
\newcommand\mydots{\multicolumn{1}{l}{\cdots}}

\begin{document}
\[
\setlength\arraycolsep{0pt}
\begin{array}{l*{6}{Cr}}
h_t     &=&ca_t&+&2c^2a_{t+1}&+&3c^3a_{t+2}&+&\mydots \\
h_{t+1} &=&    & &ca_{t+1}   &+&2c^2a_{t+2}&+&3c^3a_{t+3}&+&\mydots \\
h_{t+2} &=&    & &           & &   ca_{t+2}&+&2c^2a_{t+3}&+&3c^3a_{t+4}&+&\mydots 
\end{array}
\]
or, if lining up the ``$c$'' coefficients matters too,
\[
\setlength\arraycolsep{0pt}
\begin{array}{l*{6}{Cr}}
h_t     &=&c\,a_t&+&2c^2a_{t+1}&+&3c^3a_{t+2}&+&\mydots \\
h_{t+1} &=&    & &c^{\phantom{1}}a_{t+1}   &+&2c^2a_{t+2}&+&3c^3a_{t+3}&+&\mydots \\
h_{t+2} &=&    & &           & &   c^{\phantom{1}}a_{t+2}&+&2c^2a_{t+3}&+&3c^3a_{t+4}&+&\mydots 
\end{array}
\]
\end{document}

Is this what you want?

You have to remember that n alignment points require 2 n – 1 &. Also, using alignat* gives you full control on the spacing of the alignment columns.

Unrelated: don't use the TeX construct $$ ... $$, use the LaTeX construct \[ ... \] (unnecessary here – use the starred version of the amsmath environments):

\documentclass[11pt]{article}
\usepackage{amsmath}

\begin{document}

\begin{alignat*}{3}
h_t &= ca_t &{}+2c^2 & a_{t+1} & {}+ 3c^3&a_{t+2}+\dotsb \\
h_{t+1} &= & c \,& a_{t+1} &{} + 2c^2&a_{t+2} + 3c^3a_{t+3}+\dotsb \\
h_{t+2} &= & & & c&a_{t+2} + 2c^2a_{t+3} + 3c^3a_{t+4}+\dotsb
\end{alignat*}

\end{document} 

A variant of Mico's answer, with more alignment points, but with easier input, taking advantage of the repetitive structure.

\documentclass{article}
\usepackage{amsmath}
\usepackage{array}

\begin{document}
\[
\setlength{\arraycolsep}{0pt}
\newcommand{\row}[1]{%
  &
  c^{\hphantom{1}} &
  a_{t\ifnum#1=0 \else+#1\fi} &
  + &
  2 &
  c^2 &
  a_{t+\number\numexpr#1+1} &
  + &
  3 &
  c^3 &
  a_{t+\number\numexpr#1+2} &
  + &
  \multicolumn{3}{l}{\dotsb}
}
\begin{array}{
  l % the left hand sides
  >{{}}c<{{}} % the equals sign
  *{5}{
    l % numeric coefficient
    l % power of c
    c % 'a' term
    >{{}}c<{{}} % plus
  }
  lll % final dots
}
h_t     &=& \row{0} \\[\jot]
h_{t+1} &=& &&&& \row{1} \\[\jot]
h_{t+2} &=& &&&&&&&& \row{2}
\end{array}
\]
\end{document}