On August 5, 41 days ago, I submitted an application for a UK visitor visa. This is the third time I have applied. The only notice I have received so far says this: "Unfortunately, the processing of your application has not been straightforward. This means that we will be unable to make a decision within our published customer service standards." What should I do?
What should I do?
Just chiming in aside the great answer provided by pnuts (to whom a tip of the hat). What you should do is wait, full stop.
UKVI, as a full Directorate within an Executive Agency, has agreed with Parliament that all visitor applications will be decided within 90 days and they will have something back to you before then or else they will have to explain it the next time the Chief Inspector comes around (that's a seriously bad thing, it can disrupt someone's career).
The main thing to be careful of in these situations is if your premise becomes arguable because of the delay. For example you applied on 5 August. If your premise was to attend the Notting Hill Festival, then they will refuse on credibility (that's a bad thing) because you want to do something that doesn't exist given that it's now September. You did not tell us your premise, or if it is in jeopardy, but if it has become arguable, it can be a problem. This would come under "change of circumstances" and you signed an agreement to keep them informed so keep your side of the agreement with exactitude and diligence.
The other thing that gets people worked up is the possibility of refund penalties for tickets and accommodation. This is the applicant's problem because the Foreign Office said not to purchase tickets or make reservations in advance of a decision. If that's an issue for you, please pose a more direct question about it using the "Ask Question" button. Experts here know how to do this with dash and elan and you will get some great answers.
Another thing people do is to get their friends and relatives in the UK to complain to their MP. This tactic has rarely, if ever, worked (I have never seen it work). But worse, doing this resets your 90 day clock (because why not?).
Getting back to pnut's answer about preparing for disappointment, well yes, these things can have bad endings when there's something askew. However advice which is never wrong is: "brass it out".
What should I do?
Prepare to be disappointed. See here for what is not an answer to your question but I believe may help with your appreciating that there is nothing else you can do.