iOS: There's not exactly a dearth of apps designed to assess your mental health, but BrainyApp comes with a good pedigree: it was developed by Alzheimer's Australia. The app surveys you to determine your overall level of "brain-heart health", and then offers a range of training games and activities to improve your score.
Alzheimer's Australia estimates that the number of people affected by dementia in Australia will rise from 267,000 today to 385,000 by 2020, which is one of the better reasons to develop an app we've encountered. Right now, it's a free download for iOS only; a spokesperson told us the organisation "may look to release an Android version soon". If you give it a try, share your score in the comments.