Whilst not being the intended purpose, the app "Day One" which is available on iOS and Mac OS is a journal/diary program that is focussed on hassle free text entry. It will sit in the menu bar on Mac OS X, and you can configure reminders for how often you would like to enter something. Th eexpection is more along the lines of twice a day etc, but you can tell it as many times as you want/need.
When the reminder is due, it will produce a small popup box under the menu bar icon that allows you to type a few words in (Markdown supported, but no fancy formatting toolbars etc) and then save them, you can postpone or ignore entries if you like.
It will collate all your entries together, sync them between other instances inc. the iOS app via Dropbox or iCloud, and you are then able to search for entries specifically, or list by day/month/year all in a timeline style like a diary. It won't do fancy stuff, like applying time or entries to specific codes etc, or produce reports and so on, but if you just need manuall adhoc review of contents it's a good fit.
This is a tricky one to solve and get meaningful data.
Firstly, what defines a "word"? Usually in a text editing program this would be defined as a series of alphanumeric characters separated by a space or a new line character. That gets pretty tricky in an OS that uses those 2 keys for other options like QuickView, and enter in save dialogues and the like.
Many apps simply do not input text in a way that would make a words per minute count useful, for example safari. Would typing URLs of filling in forms provide any meaningful data, or would it just trash the more useful bits of your statistics etc.
You may be better served looking for a writing app that does this (such as this one for example - others exist) and is able to understand when it is actively being used, and when it is minimised etc, so that you are only capturing the info when you are actually doing something worth of monitoring.
To help in this there is a great tool called QuickCursor that will allow you to switch any text input field to the text editor of your choice. That way once you have found a program that you like for typing in that offers a WPM count feature, you can then use that program for all your typing by switching to it with a keyboard shortcut from any place where you need to type, including email/web forms etc, and then switching back and taking the contents with you.
Having said all the above, there is an app called TypingStats that will just sit on your menu bar and take systemwide statistics, on the proviso that it only actually guesses at your WPM by counting every 5th keystroke as a "word" and then doing the maths to give you an approximation.
Best Answer
I believe Mint.com does exactly that: https://www.mint.com/how-it-works/accounts/
They have a website as well as iPhone and iPad apps.