A regular expression for strings with at most one triple of adjacent zeroes
It's hard to say for sure precisely what is meant by "triple of adjacent 0s". My best guess would be that it means the substring 000. In this case, an easy way to get a regular expression is to find an expression for strings with no such substring, and a regular expression for strings with exactly one such substring, and then take the union.

No such substring: $r = (1 + 01 + 001 )^*(\epsilon + 0 + 00)$ should do it.

Exactly one such substring: $s = (1 + 01 + 001 )^*000(1 + 10 + 100 )^*$ looks promising, but you should check it to be sure.

If my interpretation is correct, your answer would be $r + s$. In fact, and again, I may be jumping the gun here, I think that this can "simplify" (?) to something like $(1 + 01 + 001 )^*(\epsilon + 0 + 00 + 000)(1 + 10 + 100 )^*$.