Effects that are caused BY the lens are perhaps less broad than people suggest. A lens is the conduit for light into the camera but is usually used to reflect some degree of reality. Depth of field effects are a known and utilised "limitation" but most other effects are usually seen as "distortions". This question seems to be about finding and using these creatively.

Reverse lenses for macro

Accentuated depth of field effects to make them visually obvious.
eg (trivially) Fingers spread out in front of face with one or other sharp.

Interfering objects well less than focal length from lens front to remove them when their presence is implied so that their apparent absence is remarkable (Tyger tyger - what, no bars).

Lens flare used to advantage - multiple reflections etc artistically placed - easy to just look awful.

Extremely wide depth of field effects to allow near and far to look adjacent (very small aperture and careful positioning). [eg birds in foreground, snowclad mountain many km away - birds much larger than normal wrt mountain but separation not obvious {don't overdo it}) See note 1 at end.

Chromatic aberration as an artistic effect. I havn;t thought that through BUT the "lovely" purple and green edges on some lenses under some edge conditions COULD be used to advantage by 'a master'[tm].
Note: turn off in camera CA correction first :-).

Distortion effects from wideangle lens and semi-wideangle at close enough range. Used for gotesque/humorous/interesting/unusual effects. SLIGHT spacing out of lens with macro ring will allow close focus distance to be reduced to accentuate these effects.
Subject posing and positioning can accentuate such effects. eg head with person leaning towards lens to further add to distortion of perspective.

Lens "additions": Shooting into curved surfaces, mirrors, unexpected or surprising reflections, security domes, curved mirrors in funhouse or used for wide views,

Bellows !!!! tilt and much more.

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Note 1: Wide focus field effects with apparent size alteration

Hyperfocal effect allows you to optimise focal depth by focusing at the geometric dcentre of the depth of field range of two objects so both are in focus. One traditional version of this places the far end of the in-focusrange at infinity and then has the close end wherever it falls, so that from that point out to infinity is all in focus.

BUT what I was describing achieves more than that.
In the example I gave, which was based on an actual photo situation I encountered, you can have close smallish object and distant very large objects both in focus with their relative size altered due to perspective and relative proximities.
While this is often used for obviously "trick" shots such as a person standing pixie like on someone else's hand or towering over a known tall building, it can be used much more subtly to increase the relative size of something in an eye pleasing but inobvious manner.

In the example case, gulls were flying over a seaside town with near perfectly conical Taranaki (2nd most perfect in world after Fujisan) some 10's of km away.
Using a 500mm lens the mountain filled much of the field of view and when gulls deigned to position themselves appropriately the resulting shot had them much larger relative to the mountain visually than would usually be experienced. They were not so vast relatively that the "artificiality" of the arrangement was obvious. (May try to find photo concerned and see if it is worth linking to. Better examples exist on web.)

A commonly seen example which we immediately know is artificial but which is pleasing to the eye to the extent that we ignore the obvious "wrongness" are images of an aicraft with the Moon behind it.

Two examples, both apparently genuine. The first - from here is about right - having the aircraft somewhat larger would be OK.

The second is overdone as the photographer notes -

It can be overdone- as here