I think it should be added to the FAQ that "guess the next number of the sequence of numbers" and similar are off-topic since this type of thread is posted quite often.
A few examples:
Find the missing term in these sequences
What is the sequence (1, 7, 2, 22, 3, 11, 4, ...)?
What Is The General Term Of These Sequences?
What is the pattern of these two sequences?
I actually disagree with this. It doesn't matter much to me, because I consider MO my home and only drop by math.SE occasionally. But, for the record, I think I should be able to ask a question like the following one. The details here are made up, but this is similar to e-mails I have sent to experts in the past:
In studying a problem in the subfactor theory, I have an encountered a construction which assigns a vector space to any finite group. Applying my construction to the symmetric groups $S_n$, I get trivial vector spaces for $S_n$ with $n \leq 8$, and then the next spaces have dimensions
$$1,\ 8,\ 45,\ 208,\ 858,\ 3276.$$
At that point, I overflow my computer's memory, but I hope to run $S_{15}$ on a more powerful machine soon. Does anyone recognize this sequence? These numbers all have small prime factors, but I can't find a multiplicative formula for them. The sequence is not in OEIS.
Key points: This is a question that actually comes from a practical problem, not a riddle. I give background information that might be useful, such as that these numbers come from symmetric groups and that there may be some sense in which there is a row of zeroes at the beginning. I show that I have done my work, by looking at the data I have and consulting standard references like OEIS.
A much better rule would be that there should be some external objective right answer, such as the computation for $S_{15}$ which I claim to be running.
PS I actually grabbed these numbers from this blogpost, and named subfactor theory based on the comment on that post by Dave Penneys. The vector spaces, symmetric groups and massive computation are fiction.
To the extent that these are ambiguous or non-mathematical problems (some are, some are not) what is needed is a separate site for puzzles.
The SO founders say that StackExchange is "like USENET 2.0". In terms of (a subset of) the former functionality of the newsgroups, the similes are
MathOverflow $\sim$ sci.math.research
Math.SE $\sim$ sci.math
and the natural home for the ambiguous guess-the-sequence problems would be another site to serve as an analogue of rec.puzzles.
At the moment the SE sites do not support crossposting, so cannot replicate the cross-pollination seen in the past between sci.math, rec.puzzles and sci.math.research. But having analogues of all three groups would be a good start.
The problem with these questions is in their formulation. I think they should be treated just like any other badly formulated question: The problem in the formulation should be pointed out and the author should be encouraged to edit the question to improve it.
The problem is not that meaningful questions cannot be asked about the continuation of finite sequences, but that, as the linked examples indicate, people asking such questions tend not to have enough mathematical education to formulate the questions well. Their questions tend to imply that there is a "correct" answer which is somehow determined by the initial terms, and the responses tend to point out this error and then throw out the baby with the bathwater by claiming that the continuation is completely arbitrary and the question entirely meaningless.
I believe the best response would be to point out the erroneous assumption and then go on to explain that a meaningful question would be whether there are any relatively simple mathematical structures that lead to a sequence with these initial terms, or whether a relatively simple algorithm can be given that produces one. Of course there will be gray areas where it's debatable whether any particular explanation of the sequence is obviously the best candidate, but the responses tend to overemphasize the grayness and size of these areas. If someone doesn't know that $1,2,4,8,16,32$ are the first few powers of two and asks a badly formulated question "how this sequence continues", then the response should acknowledge that there is an obvious best candidate for generating these values by a simple prescription, namely that they are the first few powers of two. The fact that not entirely objective considerations, such as what operations are considered simple, enter into such judgements shouldn't lead to the conclusion that they are entirely meaningless. We also accept other not entirely objective questions, such as what is the most natural or productive way to set up a certain problem or definition, or whether there is a more elegant solution to a problem than some known solution. Mathematics isn't just about the white of completely determined sequences and the black of arbitrary continuations; questions such as "what might be the regularity behind these numbers that I observed" also fall within its domain.
A corollary to this is that I don't think such questions should generally be closed as "off topic" or "not a real question"; this should only be done if it cannot plausibly be argued that there is a best candidate (or a small set of best candidates) to explain the regularity.
