Which version of Turbox do I need to purchase to help me get the max refund for single member LLC started in 2020?

If you are self-employed then you need online Self-Employed or any version of the CD/download.  You will need to prepare a Schedule C--which can be done in Self-Employed or any desktop software.  

https://ttlc.intuit.com/questions/2903027-how-do-i-report-income-from-self-employment

https://ttlc.intuit.com/community/self-employed/help/what-is-the-self-employment-tax/00/25922

https://ttlc.intuit.com/questions/2902389-why-am-i-paying-self-employment-tax

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https://ttlc.intuit.com/questions/3398950-what-self-employed-expenses-can-i-deduct

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Is your Single Member LLC a S corp?  Then you need Turbo Tax Business for it in addition to a personal program.

If it's not an S corp. it is a disregarded entity and you file it on Schedule C in your personal return.  You will need to use the Online Self Employed version or any Desktop program but you will get the most help in Desktop Home & Business.

If you are a single member LLC that has NOT incorporated then for tax purposes your business is a disregarded entity and you report your self-employment income and expenses on a Schedule C.  The Schedule C is included with and is part of your personal tax return Form 1040.

If you choose to use the TurboTax desktop CD/Download editions, then any edition can be used but the Home & Business edition is recommended - https://turbotax.intuit.com/personal-taxes/cd-download/

If you choose to use the TurboTax online editions and are reporting business expenses then you must use the Self-Employed edition - https://turbotax.intuit.com/personal-taxes/online/

However, if you did incorporate your business to either an S Corporation or a C Corporation then you must use the TurboTax Business Edition for your business tax return - https://turbotax.intuit.com/small-business-taxes/

The Business Edition can only be installed on a Windows based personal computer and the Business Edition cannot be used to complete a personal tax return, Form 1040.