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I think image sliders can make for a very slick home page, but it seems that Joomla has so many to choose from. What are the best, and what are their good and bad points?
Free or paid is fine.
I'm not a fan of sliders in general, however if you do need to use one with Joomla (if a client requests it, and is adamant that they want a slider for example), then Smart Slider 3 is the only one that I'd recommend.
The extension is not exclusively a Joomla extension - they offer WordPress, Magento, and a vanilla HTML version (you choose the version when you decide to download / buy, their site is not actually very clear about it until that point).
It takes a little use to get familiar with how they work, but it's worth the effort. They offer a whole load of pre-built samples, which are probably the quickest way to get up and running for new users.
Pricing is $25 for a single domain, $100 for 10, or $200 for unlimited.