Well this is a little tricky question. It depends on the country regulations the server you are connecting is located.
By definition almost every VPN service should not keep logs of their customers. After all this is what their selling. In fact the privacy protection is taken very seriously by VPN service providers because this is the essence of their service and even a single failure may ruin their business.
However, like @innogen noted, every company first and foremost is obliged to comply with the law. Thus, you should consider that your privacy is protected to the extent allowed by law and in cases of serious crimes and terrorism any company is obliged to cooperate with the law enforcement representatives.
That said, these services are good choice for the money:
https://boxpn.com/
https://anonine.com/en/
https://vpntunnel.com/
