June 2018 my pap came back as LSIL/CIN but colposcopy bx was benign. I asked my physician if this meant I had HPV and she said "No signs of HPV," so I thought I was clear. They did not HPV genotype the June 2018 pap specimen. Had another pap June 2019 and it comes back as ASCUS and High Risk HPV was detected but not types 16 and 18, and they don't result out a specific genotype beyond those two. I decided to finish my third round of the Gardasil vaccine June 2019. Negative for all other STIs. Three months ago I noticed a genital wart on myself and my boyfriend noticed he had a few warts about a month ago, which I understand is a separate issue. 

My question is, what is my risk factor for developing a cancer or other complications from the other HPV genotypes? What is my boyfriend's risk factor? (We're under the strong assumption any HPV would have came from me for various reasons). How effective is two out of three shots of the Gardasil vaccine? 

Welcome to the Forum.  I'll be glad to help.  Let me first congratulate you on your decision to complete your course of HPV vaccine.  While there is close to full protection following two injections, the 3rd shot may be associated with a more lasting protective effect.  The risk that your "high risk" HPV infection will progress to cancer is very low.  Over 98-99% of HPV infections resolve on their own without therapy over a period of about two years for most infection.  ASCUS likewise, when associated with HPV typically resolves over time- the best thing for you to do at this time is to simply get regular PAP smears with HPV testing.  Unless the PAP smear progresses or the HPV persists, there is typically little reason for any other intervention. 

If anything your BF's risk is lower than yours.  he has most certainly been exposed but at this time, there is little he needs to do. Should an abnormal skin lesions of any sort appear on his penis, it would be best for him to be seen by a dermatologist.

It sounds like you both also have genital warts which are typically more of a nuisance than anything else and rarely progress to anything more serious.  

I hope this reply provides answers to your questions.  If any of this is unclear or I missed your point, please use your up to two follow-ups for clarification.   EWH