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Green hydrogen, produced using renewables through electrolysis, can be used to reduce emissions in the hard-to-abate industrial sector. Efficient production and large-scale deployment require storage to mitigate electrolyzer degradation and ensure stable hydrogen supply. This paper explores the impacts and trade-offs of battery and hydrogen storage in off-grid wind-to-hydrogen systems, considering degradation of batteries and electrolyzers. Utilizing an optimization model, we examine system performance and costs over a wide range of storage capacities and wind profiles. Our results show that batteries smooth short-term fluctuations and minimize electrolyzer degradation but can experience significant degradation resulting from frequent charge/discharge cycles. Conversely, hydrogen storage provides long-term energy buffering, essential for sustained hydrogen production, but can increase electrolyzer cycling and degradation. Combining battery and hydrogen storage enhances system reliability, reduces component degradation, and reduces operational costs. This highlights the importance of strategic storage investments to improve the performance and costs of green hydrogen systems.