yyyy-MM-dd or die.
I remember having to fill a Canadian form (govt.) that had three date formats(in the same form), i.e.:
dd-mm-yy
mm-dd-yy
yy-mm-dd
UK here, and unless otherwise specified I write dates in yyyy-mm-dd. That order just makes sense: it is unambiguous (well, usually, see below) and sorts nicely in string form.
One extra annoyance I've experienced is SQL Server: if your user language setting is set to "English (British)" instead of just "English" (and by "English" they mean "use American standards where there is a choice"), as well as switching which way it interprets strings of the form NN-NN-NNNN it also switches how NNNN-NN-NN is read so it expected YYYY-DD-MM. This makes no useful sense at all, as far as I know no one in the world uses YYYY-DD-MM for anything... (of course what should be done for string->date conversion is explicitly calling CONVERT() with the code for ISO8601 instead of letting it guess your string format, that was it always gets it right, but in code not controlled by us there are occurrences of just throwing the string at CAST() or letting it be implicitly cast and letting SQL Server guess the format)
I can get this right every time without learning every date parrot fashion, so there must be a way to tell.
I'm not sure how largest -> smallest component (exclusive of any more fine grained units) is any more logical than smallest -> largest.
yyyy-mm-dd is only more unambiguous (as opposed to internally consistent) than either dd-mm-yyyy or mm-dd-yyyy because there is no other common 4-2-2 digit combination, while both of 2-2-4 digit combinations are common enough that ambiguity, especially with an international audience, is certain. yyyy-MMM-dd (with month name or name abbreviations) is probably the least ambiguous (even slightly better than ISO 8601 on that point), but its also language-dependent, which is a big downside, and doesn't sort well as a string (which is a downside in automated applications, though perhaps not as much for human reading).
ISO 8601 isn't perfect, its just better than most of the alternatives and adopted as an international standard.
yyyy-mm-dd has the additional benefit of being sortable as text, so it's better than dd-mm-yyyy even if you ignore the obvious ambiguity.
Yeah, if only there was an international standard that addressed that...