Pretty much everything. The service has never really worked properly and is in my opinion still in Beta. Calls get dropped all the time, even when the WiFi signal is strong. I was promised that the iOS app would ring when I received calls, but despite updating the app multiple times while they tried to figure out the problem, it never rang. Not once. Which means that every incoming call I had when I was away from my laptop (it rang on the laptop occasionally) was missed, costing me untold amounts of lost business and ruining my responsiveness reputation with existing clients.
This was just the beginning, though. What really bothers me about this company is that instead of addressing the very serious shortcomings in their service, seemed more interesting in fiddling with their business model to make the numbers add up for them and string out their venture capital a few more months. Having gone through a decision process to take on their service at the advertised price, they wrote to me saying that they had decided to retire the plan that I'd signed up and that by default I would be transferred to a significantly more expensive plan. I had already committed to changing the phone number on the website and all printed materials so this was incredibly inconvenient.
The email contained some trite explanation about being better able to serve me, that's why they'd put the price up. Well, it didn't better serve me because, as mentioned, it didn't even work, and because being abroad and deeply into a project I didn't have time to figure out an alternative.
When I finally got to porting my number out from Aircall, their incompetence was apparent. They had told me that they did not know who they had porting agreements with so I had to call around and find a provider by trial and error.
I had no choice but to remain with them during the port (which will take two weeks) and they have refused to refund a single penny of the many hundreds of dollars I have paid. Did I mention that the service doesn't even work. Requests to their support department result in my receiving numerous bullet-pointed summaries of the situation. So I will conclude this with my own:
The service doesn't work and you probably won't receive incoming calls on the app
They believe they can change their pricing on a whim
Once they have you into a contract they will do everything they can to squeeze as much money out of you as possible