I had the same trouble yesterday. IBrowse2.3 was doing something that ate limitless amounts of memory due to some glitch and it left me with a few bytes of ChipMem to save work. With such little memory you can't open a requestor - so you're done for.
MultiCX allows you to define a 'Ctrl-Alt F' for flushing memory but you'll rarely gain more than a few K.
Is there no way of leaving a browser window open and having that close/iconify when memory is tight, or even filling Ram Disk: with a large dummy file, deleting it when required?
I like this program:
ftp://de.aminet.net/pub/aminet/util/moni/ShowMem.lha
