Without modification, is the built-in pop-up flash EVER appropriate?

Sure. If you're in a situation where documenting the events in front of you is more important than making a photograph which looks nice, there's sometimes just no other way.

You can open the aperture, crank up the ISO and drag the shutter as long as possible, but if it's dark, it's dark. You don't always have control over that, and sometimes you just need the image.

I would, however, suggest that adding an external flash (and knowledge of the basics of using it) to your repertoire is worthwhile. You will almost certainly get a greater improvement in results than you would in buying a shiny new lens or this year's hot new camera body. Then, you might be able to get the image and have it look great too.

On most DSLRs you can use it, there is a "fill" option in the menus and it is great for fill flash.

Also, I've wrapped mine in a cardboard sleeve so that the light from the flash does not go forward, it just goes up. This can be used to trigger optical slave flashes. Works great if you can't afford radio triggers and don't want to trip over a PC-sync cable.

I am pretty sure that both Canon and Nikon flashes can also be used to command a TTL external strobe. In this case the flash goes off to communicate with the remote flashes, but does not go off when the shutter if open.

Webpage with examples at bottom of this post:

"Real photographers" [tm] would die rather than use a pop up flash. They'd rather have a large piece of extra equipment on top of or remotely linked to the camera at all times to allow variation in lighting level, multi axis bounce, HSS features, diffuser and reflector options, power variation, exposure method control (ADI, TTL, ...) and more.

BUT

Personally, I find having a pop-up flash IMMENSELY useful and valuable.
By all means make every effort to have a better alternative available, but the popup flash certainly has its place.

Your conditions list is tighter than reality would usually dictate is necessary - it will usually be possible to alter power and it will usually be possible to use some sort of ad-hoc reflector or diffuser OR to carry a small commercial or premade home-built unit. But, even with your limitations, it still has its place.

At the end I've added links to a number of sites carrying DIY and commercial diffusers for popup flashes. Out of these ideas it should be possible to formulate a design that you can whip up almost anywhere if needed. A4 copier paper is often available and liable to be useful. River mud and papyrus may pose more of a challenge.

If necessary you can use a sheet of paper or a (clean ideally) handkerchief to diffuse light. You can make a foldable light reflector which fits in a shirt pocket and provides a wide area of reduced intensity illumination.

A small "pocket mirror" will allow very flexible flash bouncing even when hand held. - I've just been photographing object in my kitchen using a hand held mirror and was agreeably surprised by the results - bounce flash. Note to self - add small mirror to 'ready-action' bag - probably a metal one.

But often enough the flash as-it-comes is useful.
You say flash power cannot be varied - which would be a rare situation as most systems offer flash/ambient balance and or manual flash control in-camera.
But even if this really isn't available, you can control the effect of flash by varying distance - a subject illuminated by eg sunlight will maintain essentially constant illuminance (and illumination) as you vary distance to subject but the flash effect will diminish with the inverse of distance squared.
So standing twice as far away and zooming in to compensate will drop flash effect by a factor of 4.

Shadow fill on faces in unavoidable sunlit conditions immensely improves impromptu portraits. If you can't move the sun or the landscape then shadow filling the faces can be an excellent 3rd choice.

I tend towards what is termed "street photography" in the course of everyday life. Th camera usually mounts a wide range walk-around lens, and a 50 mm and 500 mm are my near constant companions. As is a 56 guide number tilt and swivel flash with "high speed sync" (allows high shutter speed sync) ... . I avoid using flash if at all possible - but if I do need it most of the time I'll use the popup while the "real flash" stays on my hip. Wedding / function / party - the 'real' flash gets most use, but even there the popup can be useful.

A point worth noting - flash cycle time of an external time may increase as battery life drops, sometimes quite suddenly. If you are in a situation requiring multiple flashes without too high a guide number you may occasionally get caught out (bad planning but it can happen). Reverting to the internal flash can save the day - and for me at least a camera battery is much faster to change than a set of flash batteries. A fully charged camera battery makes the popup flash about as responsive in cycle time as my external flashes - especially if their batteries are low.

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Some DIY and commercial popup flash light modifiers:

Here is a nice discussion that looks at pros and cons .

Here's a use for olde world white 35mm film containers as flash diffusers BUT you could do similarly with paper. (Such containers may by now be filed away with the hens' teeth.

Here's a DIY model using vacuum bag material - but you could do similar on the fly with many materials and a paper clip of few. With and without examples given.

Some more variants some very ugly :-)

Bounce flash accessory. You could DIY with a cosmetic mirror :-) Your popup flash doesn’t have to suck

And more

Gary Fong add on looks good but too dear.

Singapore site - many versions for external and popup flash. Some good prices.

Looks weird - has some bounce component - may even work. Or not.

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ADDED:

SOME EXAMPLES:

OK. Here we go. I just KNOW this is going to hurt me more than it hurts you.

These are chosen from my "Random Maybes" folder on the basis that a flash was used and I THINK it was the internal popup in each case. In a few cases it may not have been. These are largely not photos that I'd present to critical scrutiny on the basis of their photographic merit (as you'll see :-) ) but are examples of a popup flash being usefully used.

Reasonably good results are achieved with the daylight fill-flash photos where there was lots of daylight and the flash has filled out the shadowed areas.

The bread makers sequence (shot in a dark alley in Urumqi in far Northwestern China) was shot in gloomy darkness. I had not expected to need a flash as I wandered the streets - but without one here I would have needed exposures of many seconds and some photos would not have worked at all. Full studio lighting, two lovely assistants to hold reflectors, snooted studio flashes etc would perhaps have given a better photographic result - but probably not - the popup flash probably gives a better atmosphere than anything "superior".

A number of the people shots have nasty reflections on facial highlights which a "proper" flash would have avoided - but better these photos than none at all - and in most cases the difference is more noticeable when about to be critically examined that usually. Photos of tea tables and interesting food "happened along the way" when fitting an external flash would be inconvenient and not overly justified.

Here y'go: Web page of Popup flash samples <- HERE - 60+ "examples" {tm}

ROLL / SCROLL - do not click or it will exit roll mode into album mode.

A few samples from the web page:

Andrea blowing bubbles - most lighting from Sol. Fill flash adds bubble high lights.
Much higher resolution version here

Uighur bread maker, Urumqi
Light level is consistent with Sony 5D popup flash
Increased evenness of light would have detracted from result.
Much higher resolution version here

Recycling: About 4pm in latish Autumn. Light getting down but still OK. Fill flash. Copy or download image for higher res version.
Flash shadows can be seen on coat from arm and cart shaft.
Much higher resolution version here

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Added April 2013:

Took this photo a few days ago on the front path of my property. Late in afternoon. Mid Autumn. Light getting low. On camera fill flash contributed substantially to light level. Acceptable? WWIK - but it seems bearable enough. What am I missing? (Answers to that question are welcome).
Larger 2048 x 1365 pixel version - copied from Facebook
Much higher resolution version here

The popup flash can definitely work if you are stuck in a tight situation, but would not recommend it if you are doing any serious work. For low light situations where you just want a memory but the shot is too dark the popup will do.

Although I would suggest go for something small and constant then the pop up flash. Constant light means you can do a bit of video work if you would like to but also shows what the shadows are going to be like. You can get small units like the Aperture Amaran AL-MX.

If you are set on not buying anything and do just want to use the pop up you can actually wrap the flash in baking paper as a small diffuser for a slightly softer light.

One word: shadow-lifting. It's much less prone to noise to lift shadows with a small flash close to the lens (to avoid conspicuous extra shadows) than to do it in post-processing. That is particularly true of outdoor scenes in bright sunlight where you have objects of interest in the shadow, making for a very high dynamic range. To raise the level of shadows without changing the character of shadows does not require a lot of power.

If you are a natural light buff, you can achieve comparable results with a hand-held reflector diverting some natural light in the right direction, but that tends to be quite more awkward to handle and is rarely carried around "just in case" like a camera-builtin flash.

Here is a shot with built-in flash (looking carefully at the shadows, you can see that there are some small much darker shadows where the flash didn't reach).