You will need ArcGIS For Server to create your own Map Services on your own network.
You can use an ArcGIS Online For Organizations account with publisher or admin privileges to create Map Services on Esri's ArcGIS Server instances.
This will all depend on your ArcGIS Online account.
What kind of account is it and is it for your organization or what? If you have access to host tiled services and Feature services, you should have access for Map Services as well, I would think. It all uses the same backened (ArcGIS Server of course).
This isn't really a Flex question at all, but it's not a complicated question either.
It's an ArcGIS Online for Organizations account. Currently in a 30-day free trial period. I have access to map services, just not dynamic map services, only tiled map services.
Thanks
Well then you answered your own question if you're sure that you can't create a MapServer Service through ArcGIS Online.
No, there is no way to create one with ArcGIS Desktop. You need access to the ArcGIS Server technology, whether that' through your own ArcGIS Server instance or through the ArcGIS Online frontend to use Esri's ArcGIS Server instances in the "cloud" (aka probably Akamai or Amazon's servers).