“Umasbi'a lechol chai ratzon – …and cause satisfaction for every living being…” At this point, attempts to parse the grammar falter. “Ratzon” can means desire, as we see in a few verses, “Retzon yerei'av ya'aseh – He performs the desires of those who feel awe for Him.” Or it could mean desirability, as in the end of the Amidah: “Yihyu leratzon imrei fi – May the utterances of my mouth be desirable.”
One possible argument in favor of the latter is that it would seem odd that in this verse Tehillim describes Hashem as fulfilling the desire of all living things, whereas a little later the praise is limited to doing the desires of those who merit it. On the other hand, one speaks of satisfying and the other of performing – two different things.
But there is no preposition linking it to the rest of the phrase, so the meaning is obscure. We also do not know whose desire or desirability is being discussed – Hashem’s or ours.
The Targum renders it, “And satisfies for every living being their desires.” Similarly, Rav Hirsch, in his siddur, has “satisfy the desire of every living thing”. The Metzudah and ArtScroll siddurim take the same approach. But that would more usually be “retzon kol chai”.
Shemos Rabbah (beginning of Ch. 24) points out that the Tehillim does not say “hunger” because, as we all know from experience, the hungers of a person are not always satisfied. Rather, it refers to Hashem’s Ratzon, that Hashem gives according to His Will. However, that is an idea more usually written “beratzon – in/through His Will.”
In his book on tefillah, Rav Schwab repeats a devar Torah said at his Sheva Berachos. Why is Ashrei is written as an alphabetic acrostic? The alef-beis structure represents the natural order. When we thank Hashem for satisfying our needs, we are speaking of his doing so through natural means. He does not support us by showering money upon us. Rather, Hashem provides us with material success. Rav Schwab therefore suggests that “ratzon” means desirability, and that we are praising Hashem as the One who bestows the desirability and charisma upon His creations necessary for succeeding at business.
Rav Kook offers a unique alternative. Man needs purpose, goals, something to strive for. Without wants, there is no concept of mission; boredom and ennui quickly set in. Rav Noach Weinberger says that man is a happiness seeker. Thomas Jefferson speaks of the “pursuit of happiness”. Happiness can even be defined as the emotion that drives a search, which is why we feel more happiness during the pursuit than once we have gotten used to having our goal. Along similar lines, in Borei Nefashos we thank Hashem for creating “the many souls and their lacks”. Hashem satisfies us by giving us desires. We thank Hashem for giving us purposeful existence, meaningful lives.
Having needs has a second advantage, in addition to the one discussed by Rav Kook. After the sin of tempting Chavah to take the forbidden fruit, the snake is punished that it go on its stomach and that its food will be dust. Rashi repeats Chazal’s question: How is this a punishment? Is it not good to have food wherever one goes? The punishment was in the implied statement. Hashem did not want to be bothered by the requests and needs of the snake. By giving us needs, Hashem pushes us to pursue a relationship with Him. This provides a nice counter-balance – Hashem supplies our needs, but not to the point that we are free of needing to have a dialogue with Him.
And then there are the interpretations in LN6595's answer.
I would like to suggest that the ambiguity is intentional. That, in fact, David and the One Who inspired him intended each of these meanings – and others of which I am unaware and I did not identify in this essay. This is what makes a tefillah rich. Each verse has layers of meaning so that even after years of three-times-daily repetition, there are still knew thoughts to inspire. Each time we say Ashrei we may be saying the same words, but the intent behind those words could be something that speaks particularly to what we wish to express to Hashem at that moment.