In 1971, I happened to be spending part of the summer in Aarhus, DK. About
50 miles (or maybe it was 50km, this can be checked, of course) in a
boarding school in a town called Uldum, there was a logic conference.  It
was called a counter-conference, since it was in opposition to an
"official" conference in England sponsored by NATO and the organizers of
this conference were protesting the military involvement.  Although
Grothendieck had essentially dropped out of mathematics about two years
earlier, he had agreed to come to this conference by way of supporting its
political orientation.  Although not a logician, I decided it was worth
driving 50 miles to hear what Grothendieck had to say.  Which it was,
because what he had to say was that the Giraud axioms for (what we now
call Grothendieck) toposes looked to him a lot like possible axioms for
set theory!  Now to me and probably to most people, they really don't look
like set theory, certainly not nearly as much as the Lawvere-Tierney
axioms for an elementary topos.  Grothendieck's point, which he made
explicitly, was that logicians ought to study topos theory with the idea
of understanding and exploiting the similarity he saw.  He was obviously
utterly unaware of the two-year old L-T axioms.  (In fact, Lawvere had
tried to tell him at the ICU in Nice, Sept. 1970, but Grothendieck claimed
not to be interested in mathematics any more.)  The story ends thusly: I
then asked him about the L-T axioms and didn't they resemble sets a lot
more than Giraud.  His reply was that he had never heard of them and
invited me to come up to the blackboard and describe them, which I did.
Actually, I gave a non-elementary version that was equivalent to Giraud's,
including completeness and generators, in order to be compatile with
Grothendieck's talk.  He commented that that was interesting and that is
the end of the story.