A couple of remarks at Surjections and equivalence relations lead me to wonder: are there any important and/or interesting examples of reflexive relations whose reflexivity is hard to prove? There are lots of cases where it's pretty trivial:
Divisibility: $(\forall x)(x=1\cdot x)$
Product ordering: If $(x_1,x_2)=(y_1,y_2)$, then $x_1=y_1$ and $x_2=y_2$, so $x_1 \le_1 y_1$ and $x_2 \le_2 y_2$, so $(x_1,x_2)\preceq(y_1,y_2)$.
Modular arithmetic: All positive numbers divide $0$, so every integer is congruent to itself modulo anything.
The law of identity immediately proves that the subset relation is reflexive.
I haven't, however, been able to think of a non-contrived, non-trivial example. Contrived: Define a relation $R$ by letting $aRb$ iff the Riemann hypothesis holds.
See the latter part of Leslie Townes' answer here.
In particular, note the statement that "it is often the case that reflexivity (when it occurs) is the most difficult property of these three to verify."