user1842: According to the https://en.bitcoin.it/wiki/Address site, it states that "most" addresses have a private key.  Here is the relavent text:

Most addresses have a "private key"
For most addresses, there is a corresponding secret number known as a private  key that is required to spend funds previously sent to an address.
  When using a Bitcoin client, private keys are typically stored in the wallet file. The private key has a special purpose - it is mathematically needed to create valid transactions that spend the funds originally sent to the address. If the private key to an address is lost (for example, in a hard drive crash, fire or other natural disaster), any associated Bitcoins are effectively lost forever.

I have always assumed that Bitcoin worked in such a way that all addresses must have a matching private key due to the public/private key encryption on which it is based.  In what cases would a bitcoin address not have a private key?

user1877: If an address merely 'looks like' an address and all other aspects of a transaction are ok then miners will include it in a block. 
This has been (ab)used to embed ASCII art into the blockchain at the cost of permanently destroying some bitcoin.
For an example with explanation see https://bitcointalk.org/index.php?topic=33618.msg420597#msg420597

1842: <p>According to the <a href="https://en.bitcoin.it/wiki/Address">https://en.bitcoin.it/wiki/Address</a> site, it states that "most" addresses have a private key.  Here is the relavent text:</p>

<blockquote>
  <p>Most addresses have a "private key"</p>
  
  <p>For most addresses, there is a corresponding secret number known as a private  key that is required to spend funds previously sent to an address.
  When using a Bitcoin client, private keys are typically stored in the wallet file. The private key has a special purpose - it is mathematically needed to create valid transactions that spend the funds originally sent to the address. If the private key to an address is lost (for example, in a hard drive crash, fire or other natural disaster), any associated Bitcoins are effectively lost forever.</p>
</blockquote>

<p>I have always assumed that Bitcoin worked in such a way that all addresses must have a matching private key due to the public/private key encryption on which it is based.  In what cases would a bitcoin address not have a private key?</p>
 1877: <p>If an address merely 'looks like' an address and all other aspects of a transaction are ok then miners will include it in a block. </p>

<p>This has been (ab)used to embed ASCII art into the blockchain at the cost of permanently destroying some bitcoin.</p>

<p>For an example with explanation see <a href="https://bitcointalk.org/index.php?topic=33618.msg420597#msg420597">https://bitcointalk.org/index.php?topic=33618.msg420597#msg420597</a></p>
