Why do acids usually completely react with bases? When a neutralisation reaction happens, for example, $ \pu{100 mol l^-1}$ of $\ce{HCl}$ with $\pu{100 mol l^-1}$ $\ce{NH_3}$, why does all of the base and acid get converted to salt? Why isn't there an equilibrium established between salt and acid-base?
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.