All that jQuery does (for experienced developers) is save oodles and oodles and oodles of time.
To say you've given jQuery some serious consideration, but haven't been in a situation where using it would save you time, then you're lying about one of those statements.
That's quite the accusation!
Bear in mind, I require the time-savings to outweigh any loss in execution efficiency, the necessary effect of using "swiss army functions." And don't forget the tendency, even for excellent developers, to drop best practices favor of shorter, "simpler" syntaxes.
Not a big deal for most applications. But, being a creature of habit, like the rest of mankind, I find it good to avoid using the shorthands, lest I become dependent on them in efficiency-critical solutions.
More importantly, don't assume anything I do requires jquery, or is even benefitted by jquery. I'm not one for sliding or fading tooltips, accordion navbars, etc. My needs are generally quite simple. Any my designs are simple. Both tendencies towards simplicity are deliberate.
That said, let's consider what I believe to be the single most useful method in jquery, and why it's often (but not always) better to avoid it. The $ method. When used with careful deliberation, it certainly saves a lot of typing, even for the basic and most commonly used scenario:
Much shorter, and arguably easier on the eyes. But, the 2nd line of code does WAY more than the first line -- advantageous in some cases, disadvantageous when you know what you're looking for. Based on a cursory read of the source, the 2nd line of code calls a method, which tests the given "object" first to see whether it's null, empty, or undefined, then to see whether it's already a DOM node, then to see whether it's "body" or something similar, then applies a series of matching and regular expressions to determine what to do with the string, determines that it's, in fact, an ID, finally calls getElementById(), wraps it, attaches some additional properties to, and returns something.
In the course of all this, the 2nd solution instantiates a minimum of 4 more local variables, performs at least 8 comparison, at least 1 fairly complex regular expression, and 3 additional method calls. And ... based on my 2nd look, there's even more that happens. I haven't studied the regex, but it appears as though our 2nd line above will fail the regex and call upon the find() to do what chould have been done to begin with:
Now let's consider something even more important than this ultimately roundabout means of grabbing a node: the tendency for developers to see a much simpler looking line of code and assume it's better to reuse that line of code than it is to cache the node, rather than to look it up again over and over and over ...
The developer who types document.getElementById(id) doesn't want to type it ever again. So, what do they do? They drop it in a local variable or a caching object, which is a very very very good thing to do! DOM calls are very very very expensive -- wrapping those DOM calls in a series of tests and methods increases the expense AND lures the developer into a false sense of efficiency. You start to think, "well, the jquery folks must know the most efficient way to do this. and the code is so small now. it must be better this way."
And that's not the case.
Now, I'm not suggesting we completely eschew $(). It's a second or two faster to type than the most basic alternative. And it's WAY faster to type than the series of loops necessary to fine an array of nodes by classname, for instance -- and then the subsequent series of loops necessary to deal with the resulting array.
But, I'm very familiar with the tendency to see the simplicity of $(id), and then to type it over and over in place of using it ONCE and caching the result. For most applications with low interactivity, it's not a notable concern. But, come time to develop a highly interactive application (web game, perhaps?), you'd best be well-accustomed to best practice -- something jquery allows us to to easily avoid, generally without a 2nd thought -- repeated DOM calls are bad enough for efficiency, wrapping them in layers on conditionals and other method calls can bring a real-time interactive application to its knees very quickly.
But, as Jeff alluded, the most notable examples of jquery use are amongst elite coders who know to cache the result of $() when it's needed more than once!