Yes. Perhaps the simplest example is: $$a_n\equiv n\;(\\!\\!\\!\\!\\!\\!\mod\\! \alpha)\quad\text{with}\quad0\leqslant a_n<\alpha\text{ for }n\in \Bbb N,$$where $\alpha^2-3\alpha+1=0$ with $\alpha>2$. A proof of this can be found in my paper "How slowly can a bounded sequence cluster?", published in _Functiones et Approximatio_ **46** (2), pp. 195--204 (2012). Another, rational, sequence with the same property is discussed in this paper and further in "A maximally separated sequence", _ibid_ **48** (1), pp. 117--22 (2013).
**Edit:** I realize that you had strict inequality ($>1$) in your question; you can get this from my sequence simply by scaling it up by any factor exceeding $1$.