What is the max number of SMS & MMS messages that the iphone can store?
If there is a limit apple has not posted that information on the web as I've not been able to find anything related to that.  I did turn up a few threads back from 2011 and the iOS 5 release that claim the SMS limit is 75,000 messages.  Whether that is accurate or not I do not know.

In my research I also found this application that you can use to download messages off your phone and onto your computer.  You could use this to archive old messages if you are concerned about message limits.  This can be used for SMS and iMessage.  Haven't tried it myself but it looks like a neat tool.

http://deciphertools.com/tm-mac2.html