Pseudo 1st order Kinetic reaction
Example from biochemistry

The Michaelis-Menten equation for enzyme kinetics is an example.

$v=\frac{k_{cat}E_0S}{K_S+S}$ where $S$ is substrate concentration and $E$ in enzyme concentration.

At low $S$ ($S\ll K_S$), the reaction is 2nd order with $v=\frac{k_{cat}}{K_S}E_0S$, i.e. first order in both enzyme and in substrate.  However at high $S$ ($S\gg K_S$), the reaction is first order only in enzyme with $v=k_{cat}E_0$, and is zero-order in substrate.  The substrate concentration doesn't have to be in molar excess to other reactants.

Example from process control

In chemical engineering, nonlinear systems such as higher-order reactions occuring in a continuous process at steady-state are often modeled using linear (which you can think of as first-order reactions) systems as an approximation for developing process control techniques.  A MATLAB documentation page has some good information.

Another example, somewhat contrived

As a last example, consider an unstable molecule $A$.  It can decompose to $B$ by two entirely separate elementary mechanisms.  The first is $\ce{A + A -&gt; B + B}$.  The second is $\ce{A -&gt; B}$.  At high concentrations of $A$, the second-order kinetics will dominate and the decomposition of $A$ will be second order.  But lower the concentration enough, and the reaction will become first order in $A$ because the rate the bimolecular reaction will be dwarfed by the unimolecular route at low $A$.