For the most part, this is the per-account "message per page" settings (renamed "messages to sync" in recent development builds).
This is how many recent messages the app syncs / keeps up to date every time it checks mail.
You may want to also visit app settings -> network and increase the "Preload while checking mail" settings. This is how much message text the app fetches when checking mail.
Please note, through, that a large "messages per page" setting will be used every time the app checks mail. A value over 250 can be expensive battery-wise.