One of the principal problems is that governments all around the world are incurring great costs in order to maintain a national health system.

incurring 是动词 incur 的现在分词形式。incur 由前缀 in-（表示"进入"）和词根 cur（来自拉丁语 currere，意为"跑"）组成，原意是"跑入某种状态"，引申为"招致、产生（成本或债务等）"。这里指政府正在"产生"或"承担"巨大成本。