What's the best way to share a daily/weekly curated list of great content you've found with your audience?
The only thing I currently do is create "Top posts this week" type of blog posts. Which isn't the right way to do this as overtime my blog would be full of such posts.
What are some other ways to share curated content with your readers?
There's a part of your question that I can understand very well. You seem to want "top posts this week" to be a part of your offering for your visitors, but then you're concerned about it filling your blog. It seems like if this is something you want, what's so bad about having a lot of them?
One way would be to spread it thinner by producing enough posts containing other information. If you post every day, one in five or one in seven isn't going to seem like overkill to most people, I'd imagine.
Alternatively, you could just reduce the frequency of your "top posts" posts. Perhaps it doesn't have to be weekly. Maybe it can be monthly. Or just sporadic. When you've got five or ten worthwhile articles, you make a post.
Of course, if you still want to do weekly posts but don't want them collecting over time, I see no real problem with deleting them when they reach a certain age. (When you make a new post, you'd delete the oldest remaining, such that you always only have, say, 5 hanging around.) This comes at the cost of breaking links to the deleted post, but just about every other imaginable way of dealing with this will have that same cost.
Depending on how much control you have over your blog framework, you could also just suppress posts from the top-post category (or whatever you call it) so that they don't really show up in the listings of your posts. That would reduce the collected clutter without having to actually remove them.
The final thing that comes to my mind is that you can simply present the links on your blog in a different format than an actual post. Perhaps they could show up in a module in the sidebar or footer.
Back to my original point though, if you think your audience will find these posts worthwhile, I don't see a strong need to worry about your blog filling with them over time. If people really do like them, there's a good bet they'll go hunting for them.