Is there any (free) program that allows the recording of text input and the speed of input? I am planning to run a study where participants will been shown questions on a computer monitor and their task is to type their responses in a text box as quickly as they can. Their input and input speed will be recorded. So any recommendation of preferably free programs is welcome. Thanks!
I imagine most software designed for creating psychological experiments will be able to do this. (e.g., EPrime, Direct RT, MediaLab, SuperLab, etc.). I've mainly used Inquisit to record responses and response times. These are all proprietary options.

You could also readily implement a trial interface with a textbox and response times in standard programming languages. For example, I've used visual basic on occasion to record data in textboxes.

Finally, as @Dan M mentions there is psychopy which is free. I have not personally used it.