Evernote - I know this one appears on everyone's list. I have gone back and forth between having, and not having, Evernote Premium. The 10 GB and advanced features are great, but paying a penny under $50 annually makes it a nice to have vs. a must have app. If you're watching your pennies, you might consider stepping it up to Evernote Plus -- which is half the price. It only has 1 GB of storage and is missing the business card scanner, PDF annotation and other capabilities available in Premium.
Whichever version you choose -- Evernote is pretty fundamental.
Perhaps Paper 3.0 "The new sticky note" will be giving Evernote a good run for it's money? Hard to know but an interesting one to check out.
Trello - An easy-to-adopt, robust (without being bloated) Project Management system. It might not be "Asana" -- but for the individual or small teams -- it's more than enough to keep everyone in the loop, collaborating and progressing. I used to rely on Wunderlist, but once I tried Trello I dropped the other best-of To-Do List apps.
Pocket - Having never really achieved Inbox Zero, in spite of the occasional attempt to clean out my thousands of messages, I started using Pocket instead of emailing myself links to articles I would like to read later. Granted, I might not ever get to the article, but at least the collection is all in one place. One challenge to better productivity is the Recommended articles feed. Easy to get distracted and off track.
Uber - It's easy to underestimate the impact this app can have on productivity, especially when you're on the road. I haven't used Lyft, but hear it is equally (if not better) efficient and an excellent option for getting around town.
Product Hunt - Depending on how you use it, this app might be the opposite of improved "productivity." I have lost hours while exploring the amazing ideas and new products highlighted in this app. The key is to finding collections that assemble lists of niche apps and resources to help you do what you do. For that reason alone you should definitely check it out!