What does "stick-at-naught" mean?

Solution 1:

It's a variant on stick at nothing

{ODO}: PHRASE

Allow nothing to deter one from achieving one's aim, however wrong or dishonest.

Here, it's used as a compound premodifier.

The verbal form is used by Thomas Ward in England's Reformation: A Poem, in Four Cantos as early as 1845 :

Besides, the king, tho' dear he buy it,

Will stick at naught to purchase quiet.

There is not a necessary implication of unscrupulousness, though 'he won't let anything defeat him' or 'indefatigable' or better still 'valiant' would remove most of the connotation of such. The backbiters exploit the negative connotation.