Both VoiceOver on OS X and NVDA on Windows have an onscreen display that "allows you to view all the text that (screen reader) is currently speaking".
Here's how you can display and then hide them:
To show or hide the panel:
- Press VO-Command-F10.
- When VoiceOver is on, open VoiceOver Utility, click the Visuals category, and then click Caption Panel.
For a description of the option to show the caption panel and other options to customize it, click the Help button (looks like a question mark) in the lower-right corner.
To show or hide the caption panel and the VoiceOver cursor, press VO-F11.
To enable the speech viewer, check the "Speech Viewer" menu item under Tools in the NVDA menu. Uncheck the menu item to disable it.
EDIT for those wondering:
- VoiceOver is preinstalled on all iOS and OS X installs and can be activated through Settings / Accessibility
- NVDA is an open source screen reader you can freely download (and contribute to, it's a small team ;) )
EDIT2 (2018): Firefox now has an Accessibility panel in its Devtools which display in a treeview the accessible name of each node. It isn't what a SR will read but it's its main source of information about a page through the OS accessibility API (browser software informs the OS which informs assistive technologies through their respective APIs).
Note: this is pretty low level which is both useful and useless depending on one's need.