The key is "would". But I can't figure out why "can" doesn't fit in the context.
The closer I climbed to top of the mountain, the more my breathing became labored. While I was climbing on a tough and narrow way, I could not think about anything except my goal ---- that I ____ become a better climber than my father.
I could not think about anything except my goal ---- that I will / would become a better climber than my father. Would (and will) is a statement of the future state. That's what a goal is.
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AlpheccaStarsI could not think about anything except my goal ---- that I will / would become a better climber than my father.Would (and will) is a statement of the future state. That's what a goal is.I can (and could) become a better climber than my father. This is just a statement of possibility. It is not really a goal.Thank you very much. I see.