The mpfr library actually initialises the mpfr_t data type with a value of NaN. It's something I sort of like - it implies (to me, anyway) that the developers, having no idea what value should be assigned initially, have quite appropriately assigned no value. It's up to the program to assign a value to the mpfr_t variable - and until that variable is assigned a value, it stays as NaN.

That, to me, makes perfect sense. If the programmer creates an mpfr_t variable, and subsequently tries to perform an operation upon it withouthaving initialised it, the operation raises an exception.

That seems to me to be directly analogous to:

my $x; 
$x *= 2;

Use of uninitialized value $x in multiplication (*) at ...
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Albeit that's a warning rather than fatal.

And equivalent to the trap that occurs when:

void something( char *x ) {
    printf( "%s\n", *x );
}

...
char *s;
something( s );
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Raising an exception if teh programmer attempt to use a NaN value in an operation means that he will know about it if it happens, but can deal with it if it is a possibility and there is something sensible that he can do.