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A lone mountain on a plain is powerful visual setup and one I would like to use in my world.
My concern at this point is that I don't want this location to be volcanically active, now or in the past.
Is it possible for such a mountain to come into existence without volcanic activity?
Any combination of other scientifically accurate, non-man made options, can be used to get there.
Could you be talking about a Inselberg or island mountain ? (https://en.wikipedia.org/wiki/Inselberg) Many of these are formed by old volcanic activity possibly underground and then revealed by erosion but are not volcanoes themselves.
Uluru the sandstone mountain in Australia would be an example. (https://en.wikipedia.org/wiki/Uluru)
Landforms like this can be aeolian (i.e. created by wind), alluvial (i.e. created by water) or glacial (i.e. created by movements of large masses of ice).
For example, there is a line of hills that run across the midwest (south of Columbus and north of Cincinnati in Ohio) that is a terminal moraine where a glacial dumped debris it scraped up when it melted at the front of the advancing glacier. Viewed end on, a terminal moraine can look like a single peak.
Craters and masses of atypical substance in a planetary crust can also be created by extraterrestrial impacts (e.g. meteors and comets). For example, suppose at a very large iron meteor hits a planet long in the past burying itself deep in the earth and then erosion by wind and water gradually erode the area around it, voila, you have a mountain. It may not be a really awesome mountain by the standards of places that have volcanic activity. But, in a place where everything else is pancake flat, it could still be pretty impressive.
As yet another possibility, you could also have a very tall, bushy tree with a deep root network the starts the foundation of the mountain that gradually collects dirt that ossifies over time.
The pile wouldn't have to be botanical either. The white cliffs of Dover are an accumulated pile of seashells that could take a mountain shape if carved by rivers on either side.
Also, while it isn't clear if you merely want no volcanic activity or really mean no tectonic activity, colliding tectonic plates can create mountains that are not volcanic:
The Himalayas, the European Alps, and the New Zealand Southern Alps are all examples of non-volcanic mountain chains caused purely by collision of continental crust at plate boundaries.
For a non-volcanic singular mountain, I believe your best bet is a glacial origin. Either through dirt, stone and other debris being pushed before the glacier and left behind when it melted, or through glaciers eroding the surrounding stone, etc. and leaving behind a singular mountain. Both of these are taught locally as the source of Mt. Sugarloaf.
Additionally, instead of trying to weave some sort of legends from thousands or millions of years ago about huge ice sheets into plot relevance, you could add other environmental evidence of glaciers into the story, such as Glacial Potholes, or old lake beds, flood plains, and random giant boulders (part 2, paragraphs 2-5).
While generally I would suggest volcanic activity such as the hotspot/tectonic movement combo that created the Hawaii island chain (only one active volcano, several never-ever-going-to-be-active-again mountains) for seemingly singular mountains, I hope this is useful given your non-volcanic stipulation.
Maybe of organic origin? Uplifted marine deposits can leave mountains made from fossilised reef organisms. There are examples in North America and Australia. Characterised by limestone cliffs but they may serve the purpose you want.
Brief description from the web about the Guadeloupe Mountains:
About 80 million years ago tectonic compression along the western margin of North America caused the region encompassing west Texas and southern New Mexico to be slowly uplifted. A transition in tectonic events 20-30 million years ago initiated the formation of steep faults along the western side of the Delaware Basin. Movement on these faults over the last 20 million years caused a long-buried portion of the Capitan Reef to rise several thousand feet above its original position. This uplifted block was then exposed to wind and rain causing the softer overlying sediments to erode, uncovering the more resistant fossil reef and forming the modern Guadalupe Mountains. Today the reef towers above the desert floor as it once loomed over the floor of the Delaware Sea 260 to 265 million years ago.
https://www.nps.gov/gumo/learn/nature/geologicformations.htm
