Hm. $x starts undefined

[0] Perl> use Devel::Peek;;
[0] Perl> Dump $x;;
SV = NULL(0x0) at 0x3c6d400
  REFCNT = 1
  FLAGS = ()
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You set $x to the maximum unsigned integer value; it becomes an IV:

[0] Perl> $x = ~0;;
[0] Perl> Dump $x;;
SV = IV(0x3c6d3f8) at 0x3c6d400
  REFCNT = 1
  FLAGS = (IOK,pIOK,IsUV)
  UV = 18446744073709551615
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You add 2 to it, it overflows the IV, so perl converts it to an NV (NB: No NaN arises from the overflow of the IV.)

[0] Perl> $x += 2;;
[0] Perl> Dump $x;;
SV = PVNV(0x3cb85e8) at 0x3c6d400
  REFCNT = 1
  FLAGS = (NOK,pNOK)
  IV = -1
  NV = 1.84467440737096e+019
  PV = 0
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Then, you raise that NV to the power 35; and the NV overflows; resulting in an NV with the floating point value of positive infinity (1.#INF) (NB:the "result to large" error):

[0] Perl> $x **=35;;
[Result too large] Perl> Dump $x;;
SV = PVNV(0x3cb85e8) at 0x3c6d400
  REFCNT = 1
  FLAGS = (NOK,pNOK)
  IV = -1
  NV = 1.#INF
  PV = 0
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You then divide the NV 1.#INF, by itself, and you produce an indeterminate floating point value (1.#IND):

[0] Perl> $x /= $x;;
[0] Perl> Dump $x;;
SV = PVNV(0x3cb85e8) at 0x3c6d400
  REFCNT = 1
  FLAGS = (NOK,pNOK)
  IV = -1
  NV = -1.#IND
  PV = 0
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The (special form of) NaN was produced by operating upon (erroneous) floating point (NV) values; not integer (IV) overflow.

NaN is a purely floating point concept, and cannot arise as a result of integer overflow.