I still cannot live without the Screaming Frog spider tool.
This tool does so many useful things, I don't have time to type them all in. But its best use is that it will spider your site as though it were Google, and give you nice exportable files so you can see where your dead outbound links are, which pages don't have analytics codes in them, your 2--, 3--, 4-- and 5-- response codes, how many clicks from the home page your internal pages are - like I said, too much to count.
It costs 99 british pounds a year which is something around $150 US, and worth every penny. There is a free version, but it'll only crawl 500 URLs. They are always adding significant improvements, and they are very open to suggestions - you can always hit them up on Twitter.
I also make heavy use of SEMRush (semrush.com, again you can have a free account but you can't really do much unless you sign up for paid, which start at $69/month. If you do this stuff for a living, or have a team, it's worth signing up for a higher level)
SEMRush is great for checking rankings from a neutral source (no personalization), competitor research for organic and paid traffic (and they just added product ads too) and keyword research and suggestions.
Those are my two "won't give up till they pry them out of my cold dead hands" tools.
I also like the Built With Chrome extension (builtwith.com for the desktop version, or get it in the chrome store) because I'm always looking to see what the competition is using to build their sites. The extension is free; they have paid services but I haven't needed any of them so far.