A user was looking for a way to control how much mail he stored on his laptop with an SSD drive. He wanted to create a rule to download mail and only keep it for a few days.
While Outlook can't do exactly that, it is possible to configure synchronization filters on each folder. With the filter applied and as the messages age, Outlook will remove them from the local cache, while leaving them in the mailbox online. If you need to see the older messages, use OWA or a second profile in online mode.
To configure a Sync filter, right click on the folder(s) and choose Properties, Synchronization tab, then click Filter. The filter dialog is the same filter dialog used by Views, Search folders, and Advanced Find, although some of the options are not available. After the filter is configured and you sync, items matching the filter will sync down, items no longer matching will be removed.
When a filter is configured, the status bar will show that a filter is enabled and give you one-click access to the filter dialog. The drawback? The settings are per folder. If you have a lot of folders, you'll need to set the filter on each folder.
I recommend using the Advanced tab and configuring a filter for messages with a Received date on or after 7 days ago.