Grobner basis methods for testing subalgebra membership are another example. You add extra variables in order to take advantage of the good features of Grobner bases for elimination orders. You can throw them out afterward.
For example: say you want to find out whether a given polynomial $f\in k[x_1,\dots,x_n]$ is in the subalgebra generated by a specific set $g_1,\dots,g_\ell \in k[x_1,\dots,x_n]$ of other polynomials, and if it is, to find a representation of $f$ in terms of the $G$$g_i$'s. Create a new polynomial ring with new indeterminates $G_1,\dots,G_\ell$ and $F$:
$$R = k[x_1,\dots,x_n,F,G_1,\dots,G_\ell]$$
Now compute a Grobner basis for the ideal
$$(g_1-G_1,\dots,g_\ell-G_\ell,f-F)$$
with respect to an elimination order in which $x_i > F > G_j$ for all $i,j$. If $f$ is a polynomial in the $g$'s, then there will be a corresponding element in the Grobner basis with the form
$$F-\text{polynomial in the }G\text{'s},$$
indicating the representation of $f$ in terms of $g$'s. If not, there will be no such element in the Grobner basis.
