Can I fix my open bite without extraction? (Photos)

I agree that with traditional braces extractions would be needed to close your open bite and align your teeth.  I'd have to examine you more closely and look at your lateral cephalometric x-ray to say for sure, but I don't think taking out four teeth is your only option.  If you are willing to undergo treatment in clear aligners with interproximal reduction (removal of small amounts of enamel in between your teeth to make them slightly narrower by ~.5mm) I think you could get a significant or complete reduction in your open bite.  Clear aligner treatment (such as with Invisalign) is excellent for closing open bites as the biomechanics of this treatment allows for intrusion of the posterior teeth, which causes your mandible to rotate forward slightly and close more in the front.  The more important question is what is the cause of your open bite?  Do you have a low tongue posture and a mouth breathing habit?  For the long term stability of this open bite closure you may need to consider myofunctional therapy to retrain your tongue to stay on the roof of your mouth.

Open bites are not common and are very difficult to correct.  In general, non extraction treatment tends to open bites, extraction treatment tends to close them.  Your particular case is such an obvious extraction case that it may have seemed like the orthodontists were a bit nonchalant about it, but it is so far over into the extractions necessary category that there is really no other option.Most orthodontists prefer to avoid extractions as extraction cases generally require more time, effort and cost to the patient. There is no avoiding it in this case.