Yes, they can, and they do (the average life of a single stem cell is on the order of days, perhaps less, depending on cell type). However there are many problems with this. First of all, human cells (or most any cells derived from multicellular organisms) tend not to grow well on their own. As a result, it’s tricky to figure out how to maintain growth for extended periods after removal from the body. It took years to determine how to make cancer cells grow on their after taken from the body, and they’re rather notorious for being hard to kill. (See also: “The Immortal Life of Henrietta Lacks”)
The second problem is that cells do tend to break down over time. Stem cells lines, in the proper conditions, can be made ‘immortal’; that is, they will continue to grow and reproduce as long as the conditions are kept constant. However, time takes it’s toll. Cells may develop mutations, for instance. In the body, these can usually be fixed through various methods based on feedback mechanisms. On a petri dish, these mechanisms either don’t exist or aren’t as efficient (for instance, no immune system to police cell behavior), and as a result cells lines can become decrepit over time, taking on far too many mutations and either becoming cancerous, actually evolving away from the original human genome it was taken from (which has actually happened for some cancer cell lines, which is a fascinating development in and of itself), or simply dying.
So, overall, yes. It is certainly possible to make stem cells make more stem cells. However, this is certainly not the only problem. Cells are incredibly complex, and subject to a whole host of signals and behaviors we don’t yet fully understand (which is good, because otherwise I’d be out of a job). As a result, while we can make immortal cell lines, it is still necessary to sometimes create newer cell lines.
And, this is ignoring the fact that for things like potential treatments, stem cells with similar or identical genetics to the patient are key to stopping rejection, and that would be very hard to impossible to accomplish with some random cell line. However, I think I’d probably rambled enough for today.
[edit]Also also, in response to the title question (which is slightly different from the one in the details), the cells that make stem cells are stem cells, so that doesn’t really make sense. And if you go all the way back to the original embryonic stem cells, the cells that make those are the gametes, aka the egg and sperm. So that doesn’t really help.