Can I connect a lipo battery to a nRF51822 ?

3.7V is the nominal voltage of a LIPO cell, the maximum voltage of most LIPO when fully charged is around 4.2V. In fact by the time you get to 3.7V or so the recommendation is usually to re-charge the battery else you'll start shortening its life.

So don't connect a LIPO directly as the source of a 3.6V chip, you're way outside the margins at that point.

get it. So what's the best kind of battery (performance & price) for nRF51822 ? thanks

most people use lithium ion in small devices or normal AA/AAA cells if the product is larger. Is there a reason you wanted to use a LIPO anyway? They are better known for short burst high current applications like toy cars or aeroplanes, not the tiny current usage of bluetooth.

No, there isn't. Just because of price and capacity.

Both LiIon and LiPo are 4.2V fully charged. If you want rechargeable battery, you'd either have to go with Lithium Manganeese, or use a very efficient external DC/DC that can tolerate 5V at input.