The correct spelling is : nerve-racking or nerve-wracking:

extremely irritating, annoying, or trying: a nerve-racking day; a nerve-racking noise.

(Dictionary.com)

Nerve-wrecking is a less common variant, probably from a spelling mistake. See Ngram

From The Vocabularist: Nerve-racking or nerve-wracking?:

The first recorded use of nerve-racking is in a letter by the poet Shelley in 1812, telling his friend he is glad to be away from the “nerve-racking and spirit-quelling metropolis”.
But “rack” – from a family of English words to do with stretching – had long been used in connection with torture, and often applied to parts of the body. In Milton’s Paradise Lost (1667) humanity’s coming afflictions include “Joint-racking Rheums” while the expression “Racking your brains” goes back at least as far as 1680.
In the early 20th Century “nerve-wracking” also began to appear – in the Times it was in an advert in 1905, and a 1910 news report on Peary’s planned polar expedition. The New York Times wrote “the air of America is invigorating and nerve-wracking” in 1908.
“Wrack” comes from another set of Germanic words with meanings shading from revenge into punishment and destruction, including “wreak” and “wreck”.
Some people prefer “nerve-wracking” because they associate it with wrecking – the wording “nervous wreck” is recorded as early as 1871.
It was always likely that rack and wrack should overlap. Wr- at the start of a word has been hard to pronounce from the time when w began to sound in Old English as it does today.