Over the past couple of years I have tried so many apps, but the ones which have stuck are:
Nova - my favourite launcher on both my phone and my tablet.
Executive Assistant - allows me to see quickly my emails, texts, calendar, tasks and missed calls all from one screen on my phone.
Pocket Informant - the best looking calendar/tasks organiser I have tested on my tablet.
Dice - has coped with every video I have thrown at it.
Flixster - I travel a lot and this is just perfect for quickly letting me know what is on at the local cinema
BBC News - perfect on my phone for giving me a quick update on the news and sport
Yahoo cricket - very quick and reliable cricket updates
SwiftKey keyboard - I use this on both tablet and phone
Firefox beta - I have tried so many browsers and this one has caused me fewest issues. It has even caused me to return to Firefox on my Macbook so that I can make full use of the bookmark and password sync.
Kingsoft Office - pretty decent at handling my work documents
CloudPrint - does what it says
Train Tickets by Cross Country - never failed me yet on UK rail timetables and ticket purchase
London Underground - has both a map and a route planner
Green Power battery saver - only used on my phone, but simple and reliable
exDialer - contacts and dialer in a nice simple interface on my phone