If you subscribe to the $25 per year iTunes Match service, then you can download any music that you've either matched or uploaded.
Anything on your iPhone
You might be carrying copies of important data right in the palm of your hand. At the very least, your iPhone probably contains a chunk of your photo and music library.
Your photos: Your photos will download automatically when you connect your iPhone to the new computer.
Ripped music: If you spent hours ripping CDs to your iTunes library and the songs remain on your phone, there's hope. Apple doesn't make it easy to restore music from your iPhone to your Mac, but it is possible. With a tool such as Ecamm Network's $30 PhoneView, you can extract data and media from your iOS device.
Just jack in your device, fire up PhoneView, select the kind of media you want to extract, and--if it's music, for example--copy it directly to your iTunes library by clicking Send To iTunes.
Calendar and contacts: What if you don't use iCloud, but you have synced your calendar and contacts to your still-functional iPhone? In that case, create an iCloud account if you don't currently have one, sign into it, and ask your iPhone to sync your contacts and calendars with iCloud. When you later want that data on your Mac, just create an iCloud account on it, configure iCloud to sync contacts and calendars, and your data will appear on your Mac in short order.
Your camera card
Do you regularly delete images from your camera card? If the answer is "no," today you're in luck. Keep those cards available, so you can download the images to your new or repaired computer.
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