My favorite one, in the sense that I am trying to really understand it for many years, is the determinantal gerbe of a locally linearly compact vector space, as described by Kapranov in "Semiinfinite symmetric powers".

I am making this Community Wiki since I don't really understand well this fascinating thing, so if anybody could make this answer better it would be great.

For a field $k$ and a topological vector space $V$ with certain property (linear local compactness, to be more precise, see below), he associates the $\textit{determinantal gerbe}$ $\operatorname{Det}(V)$. This is a $k^*$-gerbe. Its objects are $\textit{determinantal theories}$ for $V$. A determinantal theory $\Delta$ assigns to each open subspace $U$ of $V$ a 1-dimensional space $\Delta(U)$, and to a pair of open subspaces $U_1\subseteq U_2$ with finite-dimensional quotient, an isomorphism
$$
\Delta_{U_1,U_2}:\Delta(U_1)\otimes\det(U_2/U_1)\to\Delta(U_2),
$$
where $\det$ of a finite-dimensional vector space is its highest exterior power.

These must satisfy a coherence condition
$$
\Delta(U_2,U_3)\circ\left(\Delta(U_1,U_2)\otimes\det(U_3/U_2)\right)=\Delta(U_1,U_3)\circ\det(U_1,U_2,U_3)
$$
for $U_1\subseteq U_2\subseteq U_3$, where $\det(U_1,U_2,U_3):\det(U_2/U_1)\otimes\det(U_3/U_2)\to\det(U_3/U_1)$ is the canonical map. Then $k^*$ acts on isomorphisms of determinantal theories and turns the groupoid of such isomorphisms into a $k^*$-gerbe.

This gerbe seems to be related to very important constructions, none of them I really understand well. As Kapranov and several other authors explain, it goes back to Tate's thesis,
who used it to describe residues of differentials in terms of traces of operators on adelic spaces. Notable further works after that, to mention just the most striking ones, include "The Infinite Wedge Representation and the Reciprocity Law for Algebraic Curves" by Arbarello, de Concini and Kac (in "Theta Functions, Bowdoin 1987", PSPM 49 (1989), Part 1, 171-190), "Central extensions and reciprocity laws" by Brylinski (Cah. Topol. Géom. Différ. Catégor., **38** (1997), 193-215), "Infinite-Dimensional Vector Bundles
in Algebraic Geometry:
An Introduction" by Drinfeld (in "The Unity of Mathematics: In Honor of the Ninetieth Birthday of I.M. Gelfand", Birkhäuser Progress in Mathematics series **244** (2007), 263-304)

**Appendix** - definition of linear local compactness.

A linearly compact vector space over a field $k$ is a Hausdorff topological vector space $V$ that has a base of neighborhoods of $0$ consisting of vector subspaces and closed affine subspaces have finite intersection property. It is linearly locally compact if it has a $0$-neighborhood base consisting of linearly compact subspaces.

bundlegerbes, which are nothing more than a choice of presentation of a gerbe on the category of (smooth) manifolds by a Lie groupoid, then there are plenty of examples: scholar.google.com.au/… My favourite would be the String group on a compact, simple, simply-connected Lie group (note that it appears this really doesn't exist in the algebraic world) $\endgroup$ – David Roberts Mar 6 '17 at 8:52