Nepenthes burkei
 

My New Nepenthes burkei developed a pitcher. It is fairly easy and fast growing! The second picture shows a new pitcher starting at the bottom edge.

Cute little pitcher! I have a couple of Nepenthes I recently purchased and they are easy and fast growing. The new pitchers are so tiny!

I had no idea that they were fast growing, I always imagined them to be slow growers. How do you grow these? Similar conditions to orchids?

Ease of cultivation and speed of growth varies from species to species. Nepenthes rajah is a notoriously difficult and slow grower. N. maxima, N. alata and N. burkei are faster and easier, as are most hybrids.
As a rule of thumb, they get Cattleya light conditions. They are potted in long fiber sphagnum moss, with bark mixed in for those in pots over 6" (15 cm) diameter. I water when the moss just starts to dry out. Temperatures vary according to species, as they come from a wide range of altitudes. N. burkei does well in average room temperatures.

You have to give them water with low TDS, and they can't go in traditional potting media because fertilizer and minerals are toxic to them. Long fiber sphagnum moss with some bark in it will keep them happy, and you want to water them just enough to keep them damp. Rainwater or distilled water is best.
I have N. sanguinea and they're fast-growing, easy plants, as long as you don't give them minerals or ferts. You feed them by putting bugs (or freeze-dried worms from the fish food area of the pet store) into their pitchers for them to digest. Great for eating up all your houseflies. They need daily misting to help boost humidity, and I don't suggest them in terrariums because they get huge. They're one of the easier neps, and are generally available online pretty cheap. I love 'em.

So Fishkeeper and I are in agreement for the most part. I confess to using tap water, but mine is spring water as I live in the country. This might not work in the city where chemicals have been added, or in a different part of the country. I can see how a small N. burkei would quickly outgrow a terrarium. I generally have most of my Nepenthes outside during the warm weather so they get enough nutrients from catching insects.

There are species from the hot, humid lowlands, and species from the very cool, humid highlands. Take care to get a species or hybrid that can take whatever are your growing conditions.