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[iffi]

But in Kingdom Hearts:

There are small green boxes which represent how many health bars you'll need to go through. This way, players don't really need to memorize colors or anything to communicate stacked health bars.
I also feel like this method is easier to implement, along with cleaner in-game.
 

This system is also used in the Touhou games, and I think it works very nicely to gauge approximately how long the boss lasts. Vindictus uses something similar, in which the health bar has a certain number next to it (such as x5, x6, etc.) that indicates how many times you have to lower the health bar to defeat the boss. The healthbar is colored red for x1, x2, and x3, orange up to x7 (I think), and green if the health is above that. (Here's a video if that didn't make any sense.)

In RPGs and other games heavily based on stats systems, I, being quite numbers-oriented, like being able to see the numbers. I like the sound of that photograph system - you don't know everything from the start, but if you take a photograph you're better prepared for the next time.

[Paint by Numbers]

I think the "multiple health bars" system fits best in games like fighting games and the Touhou games where there's a sort of arms race of increasingly more difficult opponents. It really lends itself to looking/giving a feeling of increasing strength and challenge.

[Muz]

Personally, I find knowledge to be a tricky thing to play with. It either improves the exploration or gets really frustrating. I wouldn't enjoy fighting "cute bunny creature" only to find that it (nearly) kills me in one hit. Or similarly fighting a creature who looks tough, but turns out to have low HP.

Eventually, I'd just give up and look for a wiki on the thing. The likelyhood of frustration is higher, not worth it unless you're targeting it with a group of people who enjoy not knowing.

This is amplified both ways if it's an online game, where people compete with/against each other. Exploration gets more fun, frustration gets much higher.

[Defsan]

I think that the depiction of the characters' stats changes according to the amount of control you have over your character. If the combat system is just pointing and clicking, then it helps having numbers displaying health and damage (and other stats), as the combat will probably consist only of stat comparisons and some random chances (for critical hits or whatever you want to implement).
If you have more control over your character (like actually jumping, moving, running, attacking), then you probably won't want to actually represent the stats as numbers, because maybe you want the player to try and kill the monster, even if he/she isn't totally sure if it's possible.
Since the second kind of combat system is more based on skill rather than stats, sometimes I'd want the player to try and kill a stronger monster, instead of thinking "well, my STR is only 30; I better go grind some more!"

[Retrogames]

Apart from RPGs (where the number-crunching essentially defines the genre), any game where statistics were originally used to simulate action and/or dynamic values can now be done via actual gameplay actions (i.e how high you can jump, hard you hit, etc.) and graphical indicators. I'm not saying numbers are bad, but depending on how you wish to present your game, you can honestly do without them.

In terms of seeing how much damage the player (or the enemy) will cause, I would consider taking AndrewFM's advice and having parts of the bar shaded to show how much damage either will cause, with one caveat; follow ezuk's advice and not reveal this information initially. Include it after you've attacked once, after you've taken a photograph, or via some kind of tool / ability the player character obtains. For example...

The first bar shows a shaded area; this is how much damage the enemy will cause if he hits the player. The SECOND bar takes that information and applies it to the entire health gauge; each hit from the enemy will take off one division of health. An enemy that causes less damage would look more like this, where the divisions are smaller;

Again, I would keep this information hidden until the player attacks / photographs / earns the ability / whatever.

I also think that labeling equipment with symbolic ratings rather than numerals can help 'de-math' your game. If you decide you'll have 5 weapons in the game, of increasing attack power, you might identify them with stars, bullets, or an appropriate symbol rather than say "+5 attack power". Obviously a *** Weapon is better than a ** Weapon, for example.

Finally, I'm not a fan of the multi-bar method Jakman4242 suggests. Having small pips representing additional health bars the player has to burn through (or "x2", "x3", etc... as iffi mentioned), at least to me, suggests a degree of tedium, even if the fight itself isn't actually tedious. To me, that sort of thing is saying "Look how much more impressive this enemy is; instead of having one health bar, it has two! That means it's twice as hard!". I'd much rather see smaller chunks taken off the same bar than see the same chunks taken off multiple bars, if that makes any sense.