I run a very, very small business from my home, and my home easily qualifies as the principle place of business, but I don't claim the home office deduction because there's no one particular place in the home where I only do business (with the possible exception of storing inventory or a particular bookshelf). Since I don't claim the home office deduction, can I claim gas mileage from my home and back (for example to the PO to mail packages or the library when I only go there to purchase books for resale not to check out books for personal pleasure).

No; commuting expenses are personal expenses You cannot deduct commuting expense no matter how far your home is from your place of work.The time you spend driving back and forth between your home and your business is considered commuting, and the expenses associated with commuting ,standard mileage or actual expenses, are not deductible as your business expense on SCh C of 1040. ONLY
Expenses for travel between your home and other work locations are deductible if your residence is your principal place of business.

As I stated in my original message, my home IS my office: "I run a very, very small business from my home."

As I stated in my original post, my home IS my office: "I run a very, very small business from my home."

Sorry. Right; as said aslongas it qualifies as your home office, then you can claim your milegag between the home office and your work locations.