What is the best combination of professions for dungeon exploring?

I think a couple of fighters, a priest and a thief are the minimum.
Another fighter and a caster maybe if you plan to clear the place.

I worry when a party tries to delve without any fighters.

massively depending on what system you're running, and what type of dungeon we're talking about.

Can't even begin to back up this comment enough. I mean, people can beat FF1 with a Thief-only party (not even Ninjas, just Thieves) so, I even take it a step further. I've always had this strong belief that party balance is whatever you can make it. So long as there are options that might work (and a little room for trial and error) there are options that will work, and maybe that's the best option because it's the most fun, or the only one your group can run effectively.

Actually, while we're at it, can we define "Best" here too?

Survive to get in and back out with whatever was the goal of going in for was.
=

There's at least one pathfinder scenario that is composed entirely of wizards...or so I've been lead to believe.

Back in the day there was an entire series of D&D modules made to run Only Clerics, Thieves, Fighters or Mages. Class specific type things. Though any 'kit' could be used. (I am partial to Swashbucklers myself or Dual or multiclasses)

I once ran a game where we were left with a Bard, Archer, and Thief. They managed to fight off an undead horde by out running them, climbing into some rafters, and then meticulously shooting them and lighting them on fire. Creativity is the key. If they'd tried to just wade through them, they'd all be dead. Similarly, if I hadn't been willing to let them try to climb a tapestry, they'd probably all be dead, too. But as it was, the Bard actually rolled stupidly well and not only took out a bunch of undead, but also the main baddie. It made lasting memories for both the player and character, but never would have been possible without such an unbalanced group. So, essentially, any mix can make for a great game so long as players and GM are willing to work to make it so. Even with a well mixed group, that doesn't ensure that they'll do well if they don't work well together, or pay attention to what's happening in the game.

I believe that for a lot of modules, they will actually give an idea of what levels/ classes suit them well, and in that case you should aim to either have those mix of professions, or make accommodations to work around any areas that you need, but aren't covered. No Thief... maybe they find a key ring on a corpse that unlocks only certain doors for them to allow them to move on. If tanking IS needed, maybe they find a magic item that makes that possible for one non-tank. Or, a healing aid. Or, a book with rune translations. Or, an NPC that happens to have a skill set that the main group doesn't that can tag along to help. I think you can make any mix a good mix with some forethought. :)

A good GM will pitch challenges suited to the party, because a good GM isn't trying to get the party killed, but trying to stretch the party and enhance their Awesome.  Having said that, a good GM also isn't going to let the party get away with just anything.  In a way, party composition is both vital and utterly unimportant dependent on a) what kind of GM you have, and b) whether you have any specified requirements for a challenge before going in.

What I mean is this: If a party is composed of nothing but spellcasters, pushing them into a challenge in which they can't use their magic at all isn't particularly cool.  If, however, the heroes know they're headed to a challenge in which magic will be problematic at best, and don't bother taking anything but casters, that's their damage - and possibly their death certificate.

From the perspective of starting a game, or of acepting new players into an established game, the GM has the right (and, in some very real ways, the responsibility) to ensure that the mix of characters is going to be suited to their campaign.  This means talking to prospective players about what mix you want.  It means telling RTJs 'No, we have enough clerics, will you consider a fighter?'  It means making clear (perhaps in a public RTJ post, the Game Info, the ad in the appropriate forum, or all three) what kind of character you're after.  If the GM knows their campaign is going to require certain roles be filled and then does nothing to make sure that happens, their campaign isn't going to be fun for anyone.

I personally think that it's more helpful to think in terms of roles than in terms of classes.  A lot of systems out there are flexible enough that more than one class can fulfil a particular role (spell damage, non-spell damage, tank, buff/debuff, healer, loremaster, social butterfly, etc).  If the party has a broad, solid range of roles covered then they should be at least fairly okay.

And as Kessa says, creativity is key.  The players may be able to swing some crazy stuff the GM hadn't even considered - as long as the GM is flexible enough to allow them to try.

Some of my fondest gaming memories involve groups that didn't have an ideal distribution of roles (I really appreciate that distinction, by the way, as most of the systems I play don't technically have classes), and still managed to succeed.  It takes a lot of inventive thinking, figuring out how to utilize the stengths of the party, whatever its composition, and also requires a GM who's willing to adapt to an unorthodox group.  But I don't think there's a universal 'best blend' that will adapt brilliantly to any scenario...it comes down to what the players are good at playing and whether they're smart enough to tackle things on THEIR terms, rather than trying to go at it in the stereotypical hack'n'slash style.

Has anyone tried bringing an Archaeologist?  :)

I agree with Facemaker.  If you have creative players then any characters they create will make your campaign a rich story to follow.

Pathfinder-specific, but this is an example of kit-bashing abilities to fill a gap.

I once ran a game for a three-person party, and none of them had any skills good for dealing with locks. One PC was a wizard specializing in Conjuration magic, though, and one of his signature abilities allowed him to snap his fingers and conjure up a simple tool (with some limitations). So I ran with that.

I encouraged him to take the Craft (locksmith) skill, to give him some background color. I then explained that he could examine a lock and, with a successful Craft check (the same DC as the lock normally used), he could conjure a key that would fit the lock. The time required to examine the lock was the same time normally used to pick it.

I also ruled that failed attempts still used a hit of this ability, as he wouldn't know it didn't work until he actually tried to use the key. This was to prevent him from spamming rolls until he succeeded.

So, same chances and time required, but a different skill. And it allowed me to put that obstacle in their path from time to time.

This is setying the Way-Back Machine a bit, but my high school D&D group had an encounter where an Assassin ended up uaving to figgt a duel with a fighter...who actually had a few levels on him.  Not impossible for the Assassin...but long odds.  He managed to re-frame the encounter on his terms...Stealthed his way around the room, knocking out torches, until there was hardly any light left, then used Blind-Fighting to engage, and any time the fighter started to gain the upper hand, the Assassin would drop back into the shadows and maneuver around to attack from a different angle.

It was still almost a fatal engagement for the Assassin...but he won.  A thief with archery skills could probably have accomplished the same thing more easily, but the party had no thief.  It was a great display of adapting the situation, figuring out how to use different skills to accomplish the same task, and we told stories about it for years (I'm actually working on adapting it to be part of astory I'm writing.)

Unfortunately, I don't play, or run Pathfinder, so I'm wholly unfamiliar with what this means, but that sounds like I said a bad thing? Generic example, of course, for a specific dungeon crawl setting, but I liked and agree with your example of better ways to handle that. :) It's true that a lot of other abilities can also be curved, or stretched to stand in for others when needed (or even when not needed!).

The classic answer for a "balanced party" is the standard four-part approach. Fighter, Thief, Cleric, Wizard. Your based are all covered, and you aren't leaving any of the "essential roles."

But that gets old. Most parties tend to muddle through, whether it's a touring band of five bards, an inquisition of clerics and paladins, a wait of a warlock and his musclebound manservant Claude, or whatever. Characters are what make this game fun. Let the dice fall where they may.

While true, not all players are creative or up for difficult situations.
I'm thinking more "the average group of new or returning" players just getting into the game.

Some new players are more creative because they haven't been conditioned by rule constraints and some old players know the game like the back of their hand.
I'm sure these players would be fine with a half decent GM.
=

Definitely agree.  I don't consider myself the best DM in the world, but I feel that a good DM can adjust to any party variety.   Sure, some things might be tougher, but it would be a shameful thing for a  DM to say "Well, you didn't have a druid for this module, so that's why you all died".

As long as you have all your basics covered in some way, that's what's important.  Do you have a way to bypass traps?  Some way to heal up? Transportation?  Depending on how modern the time period, someone who can handle mechanical, electronic, and /or computer problems? Even if it's NPCs.

Using NPCs to save the party is a very slippery slope that is best avoided.
=

Matter of opinion.  I've been in several games where NPCs were an integral part of the group.  They didn't make any command decisions, they weren't party leaders, but they filled roles that none of the players wanted.

It's a slippery slope, yes...but I wouldn't go so far as to say it's best avoided.

DMPCs become problematic when they're allowed to overshadow the PCs. So long as they're not infallible, miles above the party's power level, or the universal skeleton-key of plot advancement, they don't generally leave a sour taste in the PCs mouth.

When in doubt, ask yourself, is the DMPC the star of this story? If your answer's yes, you might have a problem.