When Microsoft Exchange mail is configured on iPhone or iPad, the default setting for "Mail days to sync" is set to 3 days.
To increase or decrease the amount of email messages displayed in your Microsoft Exchange mail accounts (Inbox, personal folders, Sent, etc.) change the "Mail days to sync" for the account.
From the Home screen tap Settings > Mail, Contacts, Calendars. Select your Microsoft Exchange account and then tap Mail days to sync and select the number of days (No Limit, 1 day, 3 days, 1 week, 2 weeks, or 1 month)
You may notice at the bottom of the "Mail, Contacts, Calendars" screen that there is a setting under "Mail" called "Show" that allows you to show from 25 to 200 recent messages. You might think that this means that you can only see up to your last 200 messages on your iPhone, but for Exchange mail accounts, this option is ignored.
Of course, the amount of e-mail you can store on your iPhone is subject to how much free space you have on your iPhone. But with only headers loaded this means a lot of mail at 2kb per header