Rolling 6 dice and getting the same number on all I was playing a game with some friends recently.  When it was my turn, I rolled 6 dice, and they all landed on 5.  I was surprised and thought, "What are the odds?  They must be incredibly low?"  So I did a calculation $(\frac16)^6$.  However, an engineer friend said that the odds of getting that roll are technically the same as any other, it's just that our brains like to see patterns.  What do you think?  What are the odds of rolling $6$ $5$'s all at one time?
 A: It is true that our brain seeks patterns but that in itself is no reason not to take these patterns seriously. We all have brains and we can reasonably predict what patterns they will seek out. If we take this as a fact of life, then I think we can arrive at a slightly different calculation than that in your post but which, I believe, better matches the situation.
The relevant question is not 'what is the probability of all dice landing on 5?' but 'what is the probability that the dice land in a pattern that I would find as exciting as (or even more exciting than) them all landing on 5?'.
Of course you know yourself better than I do, but a reasonable guess is that patterns that qualify as as exciting as (or more exciting than) 5, 5, 5, 5, 5, 5 include all six patterns
1, 1, 1, 1, 1, 1;
2, 2, 2, 2, 2, 2;
3, 3, 3, 3, 3, 3;
4, 4, 4, 4, 4, 4;
5, 5, 5, 5, 5, 5;
6, 6, 6, 6, 6, 6;
and no others.
This makes the probability you 'should' be asking about $6*(1/6)^6 = (1/6)^5$ as noted in the comments by Thomas.

Now there is an interesting philosophical issue here. I said this probability better matches the situation than that in the other answer, even if as a calculation that calculation is of course also entirely correct.
What I mean by better matching the situation is that you intuitively felt that the probability of experiencing what you experienced was small and when we compute the probability of you experiencing what you experienced (under a reasonable model of how your brain works, as I did in the current answer above the line) this computation does indeed lead to a small number ($(1/6)^5$) confirming that your intuition was correct.
But... in the above calculation I talked about of the event that the dice fell into a pattern that you found at least as exciting as them landing on 5, 5, 5, 5, 5, 5 without going into the reason of why you found this pattern exciting. From reading your post, someone could get the impression that you found the pattern 5, 5, 5, 5, 5, 5 exciting because you (rightly or wrongly) felt that obtaining this pattern had very low probability. It seems we are running into some sort of chicken-and-egg problem here.
If we make a different model of your brain, one where you find dice patterns interesting as soon as they have low probability of occurring, the whole thing collapses. Following your friend and the other answer, there are many patterns with only slightly larger but still small probability. Now if indeed you would find them all equally exciting, my calculation above would not match your intuition at all since the probability of getting any dice pattern that by itself has low probability of occurring and 'hence' be exciting according to this model, is pretty much 1. This is what your friend is hinting at.
So what is going on here?
I stand by my claim that computing the probability of you seeing a dice pattern that you find interesting is the right computation to do.
The question then becomes: do we model your brain as finding dice patterns interesting as soon as they have low probability of occurring or do we model your brain as finding dice patterns interesting only when they have low probability of occurring and some other appealing property that makes them stand out? (Like all dice showing the actual same number rather than just numbers with the same probability of occurring)
As I wrote above, working under the first model yields a probability of 1 (or something close to it, depending on the details of the model). Working under the second model yields the small probability of $(1/6)^5$ which (by being small) better matches your real life experience of feeling that you just watched a low-probability event.
This better match to the real world is a justification for choosing the second model. But as I wrote at the beginning of the post: even before throwing the dice we could have picked the second model over the first, because we all have brains and know how brains work.
A: Ah, your friend is not quite correct!---at least, not in the usual way that dice games are played. Consider rolling five 5s and one 6. Any of the six dice can be the 6. So there are six ways of doing this.So it's six times more likely. Let me explain with an example.
If you were to have six differently coloured dice, then indeed "all six come up 5" is the same probability as "red = 1, blue = 2, green = 3, orange = 4, black = 5, white = 6". After all, it's just "red = 5, blue = 5, green = 5, orange = 5, black = 5, white = 5". But, usually, dice are indistinguishable from each other; alternatively, if they are distinguishable, eg by colour, it doesn't matter for the game.
So "one 6 and five 5s" can be any of the following:

*

*red = 6, blue = 5, green = 5, orange = 5, black = 5, white = 5;


*red = 5, blue = 6, green = 5, orange = 5, black = 5, white = 5;


*red = 5, blue = 5, green = 6, orange = 5, black = 5, white = 5;


*red = 5, blue = 5, green = 5, orange = 6, black = 5, white = 5;


*red = 5, blue = 5, green = 5, orange = 5, black = 6, white = 5;


*red = 5, blue = 5, green = 5, orange = 5, black = 5, white = 6.
Each of these has probability $1/6^6$, as does "six 5s". So,
$$
\Pr(\text{six 5s}) = 1/6^6
\quad\text{but}\quad
\Pr(\text{five 5s and one 6}) = 5/6^6.
$$
