Q: My computer fried.  Does itunes remember my play list?

No.

It has always been very basic to always maintain a backup copy of your computer.  Have you failed to do this?

It isn't a racket.  You simply don't understand computer ownership, namely having to maintain a backup of files in case of computer failure (and any computer can fail).  Frankly, people would probably be horrified if Apple kept a copy of people's playlists and consider it a great invasion of privacy.

What you can do, but was not asked in your question, is re-download most media you have purchased from Apple.  For other things it is your responsibility. 

drandersen wrote:

 

My computer fried.  Does Apple's itunes store my playlists?

Yes.

Do you have a backup of your computer with iTunes on it?

drandersen wrote:

 

By re-down loading, do you mean re-purchase?

No.

You don't need to repurchase anything. Just redownload your past purchases.

See this -> iTunes - Download past purchases

Downloading (using iOS or computer) past purchases from the App Store, iBookstore, and iTunes Store - http://support.apple.com/kb/ht2519 - enabled with iTunes 10.3 and newer; not all media formats are available in all countries (see: iTunes in the Cloud, iTunes Match, and iTunes Radio availability by country - http://support.apple.com/kb/HT5085); apps, books (not audiobooks), music, t.v. shows, and movies (some - not all studios have permitted this).  Downloading previously purchased movies and TV shows requires iTunes 10.6 or later.  Discontinued items not available. For items not included in the iCloud list (e.g., ringtones, audiobooks), or locations or computer systems where iCloud is not (yet?) available, you only get one download per fee paid.  Apple recommends,  "... back up your iTunes library. " (http://support.apple.com/kb/ht2519).

Additional tip at https://discussions.apple.com/message/24176756#24176756

  Select the iTunes Store on the left side of iTunes if you have the sidebar displayed, or with the button at the top right.

  Click on Purchased on the right side under Quick Links (see https://discussions.apple.com/message/26364186#26364186)

  You can re-download your available previous purchases.