Which tax package will allow me to file 3 Fed and 3 State (same) returns for my household?

[TomK100]

Federal returns: Although you can paper-file as many federal returns as you want using the CD/Download software, only five (5) of those can be e-filed per IRS regulations.

State returns: If you have more than one state attached to your federal return, up to three (3) of them can be e-filed. Keep in mind that many nonresident state returns can't be e-filed anyway, so if you're doing more than one state, focus on e-filing the resident and part-year resident return(s).

[VolvoGirl]

Only the Desktop CD/Download program installed on your computer can do multiple returns.  Deluxe and above comes with one free state program download.  Buy it here,

https://turbotax.intuit.com/personal-taxes/cd-download/ 

Online is only good for 1 return per account.  But you can use the same email address for 5 accounts.   How to start another return in the Online version

https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-start-another-return-in-turbotax-onli...

If you use the Online version maybe some returns can use the Free Edition or Free File website.