In physics, the canonical example is the case of gauge theories. There are different levels of sophistication:
Classical electromagnetism: here the "physical" variables are the components of the two form $F$ (the electromagnetic field). In principle, one may formulate the theory purely in terms of $F$, with no extra camels. On the other hand, the general analysis greatly simplifies if one introduces extra variables in the form of a one form $A$, such that $F=\mathrm dA$. As usual, the differential form $A$ is defined up to an arbitrary exact form. General field configurations (the solutions of the equations of motion), when written in terms of $A$, depend on one arbitrary function, reflecting the presence of extra "unphysical" degrees of freedom.
Classical Yang-Mills: here we have several two-forms $F$ (living in the adjoint representation of a certain Lie Group) such that $F=\mathrm dA+A\wedge A$. Here, and unlike before, it is impossible to formulate the theory purely in terms of $F$, and therefore one is forced to introduce the one forms $A$. As before, these are not uniquely determined given $F$, and therefore general field configurations depend on several arbitrary functions.
Quantum Yang-Mills: in order to have a manageable and consistent theory, one is obliged to introduce more "unphysical variables", i.e., more extra fields. These are the so-called Faddeev-Popov ghosts, and they greatly simplify the general formalism but, once again, they are auxiliary, unobservable fields. They do not affect measurable predictions.
Other quantum gauge theories: the most general and systematic approach to the quantisation of gauge theories that we know of as of today is the Batalin-Vilkovisky formalism. In this formalism, one introduces one auxiliary variable for each field that one is working with. That is, the first step towards the quantisation of the theory is to duplicate the number of variables (which include the gauge fields themselves, as well as any Faddeev-Popov ghost, or any other auxiliary field that is present). These variables are eventually eliminated, but their presence greatly simplifies the general analysis.
Other examples in physics:
Stückelberg fields, which are extra fields that are introduced to maintain gauge invariance in massive theories. In the same vein, the Goldstone-like components of the Higgs field.
Pauli-Villars fields, which are introduced to ensure convergence of certain integrals. They are eventually eliminated (after making sure that there are no measurable divergences).
Inonu-Wigner contraction: sometimes, one may understand a certain group as the contraction of a larger group. If the larger group is simpler than the contracted one, one may use this information to simplify the analysis of the contracted group. Such is the case of the Poincaré Group, which is the contraction of $\mathrm{SO}(4,1)$ (the latter being semi-simple, and the former not). The explicit contraction allows us to obtain, for example, the rank and Casimirs of Poincaré in terms of those of $\mathrm{SO}(4,1)$.
The method of image charges, which is a strategy to solve certain problems in electrostatics by introducing extra "fictitious" point-charges, and exploiting the symmetry of the resulting configuration. One should note that this technique can be used to obtain certain Green functions provided the domain is simple enough.
Finally, I'm surprised no one has mentioned the standard trick to calculate the Gaussian integral $$ \int\mathrm e^{-x^2}\mathrm dx=\sqrt{x} $$$$ \int\mathrm e^{-x^2}\mathrm dx=\sqrt{\pi} $$ by introducing a second coordinate $y$, such that $$ \left(\int\mathrm e^{-x^2}\mathrm dx\right)^2=\int\mathrm e^{-r^2}\mathrm dx\,\mathrm dy $$ and changing to polar coordinates.
