I've watched a lot of episodes in Stargate series. I'm not really sure, but whenever they mention ascended beings, they are always the ancients, the Ori and Anubis, which was half-ascended somehow. Why didn't the other advanced species ascend? For example, I remember an episode in which the Asgard were dying due to the weakening caused by cloning, that seemed like a great opportunity to ascend, I guess.
In the SG-1 series finale:
We find out that the Asgard can't Ascend due to the way they intervened in their own natural evolution.
Addendum: In the non-canon Atlantis novel Unascended (set after season 5), there is at least one Asgard who acended before the Asgard starting using cloning. While not canon in the strictest sense, it does show that other people have been asking the same question.
This answer will be full of spoilers. Rather than tag the whole thing, since I don't know how much of SG-1 or the other series you've seen, here it is:
As the other answer states, the Asgard are revealed in the last episode of SG-1 to be unable to Ascend, due to interfering in their own evolution millenia earlier. While this enabled them to live much longer lives than their unimpeded biological evolution would have allowed, they inadvertently trapped themselves in an evolutionary dead-end, which they were unable to escape, despite numerous attempts. It is unknown if the "Lost Tribe" of Asgard in the Pegasus galaxy, the Vanir, have quite hit this cul-de-sac.
The Ancients also didn't come by Ascension naturally, but are shown in Stargate: Atlantis to have been researching Ascension for some time before they were forced out of the Pegasus galaxy. The Ori presumably did the same. Anubis was only able to Ascend by tricking Oma Desala into helping him do so, but once he did he retained enough knowledge of the process that he could create clones of himself capable of Ascension, in a plan to fight a war with the Ancients. He was trapped in a never-ending struggle with Oma Desala before his prototype warrior, Khalek, could be completed; Khalek did, however, retain enough of his creator's knowledge to attempt to repeat the process on his own, but was killed by Daniel Jackson and Cameron Mitchell before he could do so.
Some humans are shown to be able to achieve Ascension, but only after being instructed by the already-Ascended, or after generations of meditation and training. The Nox are never shown to have either achieved or not achieved Ascension, but their pacifist nature might preclude them from the necessary experiments on other life-forms the Ancients seemed to need to reach that level. The Furlings may well have wiped out by the Goa'uld, as seen in Harry Maybourne's second-last appearance on SG-1.