give up 和 give up on 都可以说，但有区别：
give up (doing) sth 意思是“停止（做）某事”“放弃（做）某事”。如：
He was forced to give up his job. 他被迫放弃自己的工作。
She refused to give up trying. 她不肯放弃尝试。
My doctor urged me to give up smoking/cigarettes. 我的医生竭力要求我戒烟。
Rescuers have not yet given up hope that more survivors will be found. 救援人员还没有放弃找到更多幸存者的希望。
He gave up his seat to an elderly woman. 他把座位让给了一位老妇人。
give up on sb / sth 意思是“对某人或某事不再抱希望或不再相信某人或某事”。如：
His teachers seem to have given up on him. 他的老师似乎不再对他抱有希望。
I’ve been in trouble so many times that my teachers had given up on me. 我惹了太多次麻烦，老师们对我都不抱有希望了。
At that point I hadn’t completely given up on the marriage. 那个时候我对这桩婚姻还没有完全放弃希望。
现在回到网友的句子：
He showed us that even a single desire, never given up on, can make for a remarkable life.
never given up on 是过去分词短语作状语，表示条件，意思是“只要永不放弃”。全句意为：他向我们表明，即使只是单单一个愿望，只要永不放弃，也可以成就非凡的人生。
