If you can avoid the need for concurrent writes to a single file, it sounds like you do not need a database to store the chat messages.
Just append the conversation to a text file (1 file per user\conversation). and have a directory/ file structure
Here's a simplified view of the file structure:
chat-1-bob.txt
201101011029, hi
201101011030, fine thanks.
chat-1-jen.txt
201101011030, how are you?
201101011035, have you spoken to bill recently?
chat-2-bob.txt
201101021200, hi
201101021222, about 12:22
chat-2-bill.txt
201101021201, Hey Bob,
201101021203, what time do you call this?
You would then only need to store the userid, conversation id (guid ?) & a reference to the file name.
I think you will find it hard to get a more simple scaleable solution.
You can use LOAD_FILE
to get the data too see: http://dev.mysql.com/doc/refman/5.0/en/string-functions.html
If you have a requirement to rebuild a conversation you will need to put a value (date time) alongside your sent chat message (in the file) to allow you to merge & sort the files, but at this point it is probably a good idea to consider using a database.