Which one is correct? “ I went to my ...”

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I feel like this is one of those situations where I reword it so I don’t mess it up. 
I went to an appointment at my doctor’s office. Haha

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It's not doctor's appointment because it would be like saying doctor is appointment.

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either doctor’s or doctor is correct. with doctor’s you have an appointment with the doctor and the doctor also has an appointment with you, therefore it is the doctor’s appointment. when using the later of the two it is also correct as saying you have a doctor appointment you are stating the type of appointment you have, an appointment with a doctor, much like saying I have a hair appointment. more than one doctor would be doctors but you’d say is as “i’m going to an appointment with my doctors” so in the order of the word “doctor” followed by the word “appointment” you can use either doctor or doctor’s but not doctors.

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I’m still not completely sure (hence why I asked the question), but I think the correct way is “doctor appointment.” I’ve been getting mixed answers on Google, but this is the one that I believe makes the most sense:

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So can someone confirm what is correct

Which one is correct? “ I went to my ...”

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