What are other elements other than time and space that are said to have dimensions? Most laypeople would say that only space can have dimensions since when we say 3d we immediately think of space, but some people also argue that time can have dimensions. What are other things that are thought to have dimensions in philosophy by some people?
Any thing with several totally different and unrelated quantifiable features can be classified and studied by treating those features as dimensions. For example, a song can have mood (passion, angry, joyful, melancholy, etc) and genre (jazz, rock country, soul, rap, etc) as its 2 different unrelated features each of which can be somehow technically ordered and thus quantifiable. Then Pandora or Spotify may use AI to map all the songs in their database and recommend one for a user based on the least distance in this 2-d feature "space" to the song already picked by the user before.
Any meaningful substance existed in philosophy usually has a complete set of many or uncountably infinitely many features (also called properties, attributes, characteristics, etc), so in this sense any real substance has its own "nd phase space" borrowed from maths jargon.
The word "dimension" has various usages. Some are of greater philosophical interest than others.
It is often used as a metaphor. For example a famous aeroplane designer once remarked that "Every aeroplane has four dimensions; length, breadth, height and politics".
In mathematics, you can have any number of spatial dimensions you want; theoretical physics can run into double figures. Some theories may occasionally add an extra time dimension - or treat even the one as an emergent property of some timeless fundamental reality.
More generally, in mathematics and engineering, degrees of freedom are sometimes referred to as dimensions. For example, a mechanical flight simulator can not only move in three orthogonal directions but can also rotate about three different axes, giving it six degrees of freedom in all; this is often referred to as full six-dimensional motion.
More generally still, anything treated using similar equations to those of geometry will be talked of as having dimensions, sometimes with the caveat that these are not spatial dimensions as such. An example is provided by linear programming, where the number of dimensions corresponds to the number of variables being optimised.
In mysticism and metaphysics, an immaterial realm is usually postulated, and this often has a duplicate set of the familiar dimensions of physical spacetime. Indeed, many such realms may be stacked up. Back in 1927, J.W. Dunne proposed an infinite series of time dimensions in An Experiment with Time, along with an infinite series of levels of the mind to go with them. Several subsequent writers either accepted or seriously considered his second level of time, as the only meaningful dimension beyond the physical, but rejected all the rest.
The borderline between speculative physical theories and unfalsifiable metaphysics can be a narrow and contentious one.