This answer is meant only if the context of such a question is some quiz or test, not if it occurs of, say a truncation of some approximated infinite series.
First arithmetical simplicity, then "translation" of the sequence to some current common knowledgde. Well, "simplicity" is a term lacking sharp bounds, so some "least square error" may be with the optimal guess...
Arithmetically: equal difference of first or second order, equal quotient. Equal quotient after removing a constant. Do the numbers have some common property, say they are squares, cubes. Possibly they are obvious moduli of some other sequence. Then I would try common sequences like primes, fibonacci..., just what is present in mind.
If no obvious answer, look at the form of the elements. Do they have only one digit? Perhaps the beginning of a well known constant.
If nothing matches so far, look at OEIS. If still nothing, it becomes harder.
But if it is so hard, then -if we are in a test- maybe the numbers represent something else, they are some transformation (like date or time information) or are numerical values for some common social construct related to required knowledge in the framework, where the test-situation occurs. Politics (years of important incidents...1,9,39,8,5,45,23,5,?? answer:49; why? (german historical knowledge required, sorry ;-) )) Economics, sports, social events... just what is up currently.
A third class are sequences, where the terms are codes, for instances code letters of the alphabet.
If all that has no success, I'd try to figure out the polynomial solution which is always possible...
An example for a non-obvious guess (illustrating some real-math-problems):
Actually, in my heuristic studies of functional and series-relations it is a problem at the heart of many problems to find approximations to rational numbers, if a function with a so-far unknown characteristic is approximated using only finitely many terms of its powerseries. It is much arbitrary to interpret some number like 2.44269504089 (of which I can assume it is only an approximation). Even if I have "the next one" 3.08136898101 and also the next one 4.00278070716, still for each one many interpretations can be found.
But if some interpretations of the three numbers have the common element log(2) with some rational number and small error , and especially, if the log(2) component has consecutively increasing powers of log(2), like (3.08136898101-2.4426950408) . log(2)~ 0.442695040954 and (4.00278070716-3.08136898101) . log(2)~ 0.638673940116 but again (4.00278070716-3.08136898101) . log(2)^2 ~ 0.442695040954 then I guess the continuation as something "obvious" and begin to try to make a proof for it - because of the argument of "arithmetical simplicity" at the beginning of my post...