If your SAP domain is example.mycompany.com, your from address would need to be [something]@example.mycompany.com (where [something] can be any alphanumeric string).
Alternatively, you could use [something]@mycompany.com or any subdomain like [something]@offers.mycompany.com providing you have have mycompany.com set up as private domain.
However, if your SAP domain is example.mycompany.com and your from address is [something]@othercompany.com, this will fail DMARC validation.
You will note that there is a new verification tool in the current release, where you can verify a domain in the Sender Profile. Refer to documentation.
However, if you do not have SAP enabled (which your title infers), you would need to have a Private Domain for the email domain that you intend to use (but bear in mind, link tracking, images, etc would use a generic exacttarget.com domain). This would need to be used with a shared IP address pool.
But please note that if you are sending more than 100k messages/month (the minimum needed to keep a dedicated IP ramped up), I would recommend you use an SAP, which will give you a private IP address and link/image branding.