Why do acids usually completely react with bases?
Acids and bases don't necessarily completely react with each other. Consider your hypothetical reaction between hydrochloric acid and ammonia. 

\begin{align}
\ce{HCl + H3N &amp;-&gt; H4N+ + Cl-}&amp;
 \text{Extent:}&amp; \approx100\%
\end{align}

The products are ammonium ion and chloride ion - the conjugate acid and conjugate base, respectively. So if the forward (above) reaction is complete, how can the reverse reaction between the conjugate acid and conjugate base be anything but insignificant? I.e. 

\begin{align}
\ce{H4N+ + Cl- &amp;&lt;&lt;=&gt; HCl +H3N}&amp;
 \text{Extent:}&amp; \approx0\%
\end{align}

An equilibrium is established whenever the extent of reaction is not complete. Another good example would be an acetic acid solution. Acetic acid and water exist in equilibrium with acetate anion and hydronium ion. So no not all acid-base reactions go to an extent of 100%. Things to consider: stoichiometry and acid and base strengths.