Since is countable, you can consider the sequence formed by all the rationals, obtaining then a sequence such that its cluster points are ALL the real numbers!
It is a concrete example!!! is countable and dense. Countable means that there exists a bijective aplication . Hence is a sequence containing ALL the rational numbers (not formed by rational numbers). Since is dense in , every real point is a cluster point of this sequence.
If you want a simple constructive formula then you could take , which oscillates slowly between –1 and +1, so that every point of the interval [–1,1] is a cluster point. That also provides an answer to the question in your other long-running thread Limit of a Sequence.