It depends enormously on your personal objectives, apart from your personal predilections. For example, if a significant goal is to advance your understanding of mathematics, then obsessing over exercises (many of which are contrived busywork in undergrad textbooks, and sometimes in grad-level textbooks in the U.S.) is a dubious investment of your personal resources.
For one thing, apart from the articiality of some of the exercises, many of them will be semi-incomprehensible if you've just read the chapter they appear after... but obvious after you've read further! A significant reason for this is that mathematics has developed with various goals in mind, so that the most important enduring concepts and facts refer to important phenomena... not just to some artificial choice of linear logical development as is the common style in textbooks.
On another hand, if you do not aim to be a professional mathematician, or if somehow you have a lot of spare time, sure, why not do whatever you want? Indeed, another common trap of studying mathematics is being too obedient about following some syllabus or textbook, as opposed to following one's own curiosity and interests. It is subtler to parse the situation that your impulse is to do all the exercises... :)
Another practical point is that, at some point, probably soon, unless you severely restrict what books you look at, there's no way you'll have time to do all the exercises in detail, even if you are a whiz-kid. There are too many, and sometimes they are prankish. For example, the "exercises" in Atiyah-MacDonald's "Commutative Algebra" (a misleadingly slim volume) are mostly "theorems" one would find in other books on the same topic.
And, then, there's the point that novices' "solutions" to difficult exercises are often severely suboptimal, even if "successful". Sure, it's good to think about issues, but, at the same time, you'll be able to approach those questions far more wisely later (if you still care, and the things haven't become completely obvious anyway!).
But certainly no one is "required" to do all the exercises, despite some propaganda on the internet. For that matter, it is probably not optimal for most peoples' circumstances and goals. Still, taking an extreme stance, maybe the world will end tomorrow, and if you want to spend the evening doing exercises that you find enchanting, I'd be the last one to try to discourage you. :)