OpenSeadragon
Openseadragon is a deep-zooming javascript client whose primary advantage is a very smooth loading and navigation experience. It grew out of a research project alongside PhotoSynth at Microsoft Labs, and its progenitor was featured in a TED talk in 2006.
The Openseadragon library ports this technology to javascript. It supports the IIIF Image API natively, and is very simple to get up and running by supplying an info.json reference as its "tileSource". This is described in a dedicated page in its documentation.
To experiment with its configuration options, try this jsfiddle.