The following are examples straight from the California Franchise Tax Board, FTB 3556 LLC Memo (publicly available with a simple Google search), as to what California will deem is "doing business" in California, thus subjecting you to the $800 minimum tax. I would suggest that you think long and hard about how your corporate and LLC setup falls in these examples. As you'll see, it doesn't take much for you and your corporation or LLC to be deemed "doing business" in California.
Example 1
Paul is a California resident and a member of a Nevada LLC. The Nevada LLC owns property in Nevada. The LLC hires a Nevada management company to collect rents and provide maintenance. Paul has the right to hire and fire the management company. He occasionally has telephone discussions with the management company regarding the property. He is ultimately responsible for the property and oversees the management company. Paul conducts business in California on behalf of the LLC. The LLC must file Form 568.
Example 2
Rachel is a California resident and member of an Oregon LLC. The Oregon LLC has a retail store in Oregon. Rachel uses a California address for the LLC's tax filings and a California accountant to prepare the LLC's tax returns. Rachel conducts business in California on behalf of the LLC. The LLC must file Form 568.
Example 3
Sara is a California resident and a member of a Texas LLC. The Texas LLC receives royalties from Texas oil wells. Sara maintains a California business bank account and secures financing in California for the LLC's Texas investments. Sara conducts business in California on behalf of the LLC. The LLC must file Form 568.