State-of-the-art approaches to partially observable planning like POMCP are based on stochastic tree search. While these approaches are computationally efficient, they may still construct search trees of considerable size, which could limit the performance due to restricted memory resources. In this paper, we propose Partially Observable Stacked Thompson Sampling (POSTS), a memory bounded approach to openloop planning in large POMDPs, which optimizes a fixed size stack of Thompson Sampling bandits. We empirically evaluate POSTS in four large benchmark problems and compare its performance with different tree-based approaches. We show that POSTS achieves competitive performance compared to tree-based open-loop planning and offers a performancememory tradeoff, making it suitable for partially observable planning with highly restricted computational and memory resources.