Does there exist any Apple hardware "Cloud Computing" solutions? I have a program which uses Mac OS X's Grand Central Dispatch queueing technology to handle multi-core programming.  However, I notice that, in the near future, Microsoft is expecting to roll out cloud-based multicore programming where you can run your parallel programs on a ridiculously large (8,000+ core) setup with a pay-by-the-hour setup.

Are there currently plans for Apple to offer similar iCloud-based "Big Computing" in the near future?  Is there already something like that which exists that I am unaware of?
No - Apple stopped selling both its enterprise storage Xserve RAID as well as its enterprise server Xserve in February 2008 and January 2011 respectively. Even the workstation-class Mac Pro has only seen incremental updates over time so the only way to get "into the cloud" with Apple as of late 2012 is by developing software that uses their iCloud API and service.

The only credible rumor of Apple product is the very vague announcement to the "Pro user community" from a short email by Tim Cook that promised that Apple are "working on something really great" for later in 2013. Whether that is going to be new (and perhaps cloud-based) hardware or software or just a re-design of what we've already seen is left to a lot of rumor and speculation at present.