To reduce the number of vertices of a mesh circle to five? I must be missing something.
Well… You just do it.
Or in GN:
Or manually:
* E , not Shift + D , sorry… Shift + D would also work though just as the last step you would fill with F , not merge by distance…
Or with Python:
import bpy
import bmesh
import math
m = bpy.data.meshes.new("Pentagon")
bm = bmesh.new()
verts = []
sides = 5
for i in range(sides):
angle = i * (2 * math.pi / sides)
verts.append(bm.verts.new((math.cos(angle), math.sin(angle), 0)))
for i in range(sides):
bm.edges.new((verts[i], verts[(i + 1) % sides]))
bm.to_mesh(m)
bm.free()
o = bpy.data.objects.new("Pentagon", m)
bpy.context.collection.objects.link(o)
bpy.context.view_layer.objects.active = o
o.select_set(True)
Any more ways? 
Text! We can do text!
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Copy paste to text object, then convert to mesh.
Obviously you can specify the number of vertices when adding the circle. How then (now) do you edit these parameters after you have moved on to something else? This used to be trivial.
You’re asking for an edit history? ->geometry nodes
Did it? (not as far as I remember, without addons)
Until geonodes came along.
I would think exactly the same way - just do it! It’s 5 verts… How hard can it be?..
I think you might be mistaking something here. It has always been like this or less trivial. We have Repeat Last Operation panel for… I don’t know… more than a decade?.. It has always been available only until another operation is performed. Obviously now it’s way more trivial because we have Geometry Nodes, where you can do pretty much whatever you can imagine, like others pointed out. What exactly are you talking about? How do you need to edit the pentagon? What’s the context here?
Having created the circle with 15 vertices and a fill-type of nothing: how to change the number of vertices and the fill type.
I don’t know of any trivial way that existed to do so. Create a new one, merge and snap?.. I don’t know. Depends on situation maybe. I am not aware of any particular universal trick or that it has ever existed from the time I am using Blender though. I think I started with 2.64…
Well, you can always merge or use checker deselect and disolve the verts to the right amount then use LoopTools addon’s Space or To Circle… I don’t think that sounds like the answer you are looking for though… 
I’ve seen tutorials, and I’ve done it. The mesh circle was created with 20 vertices and no fill. Now, I want to change the number of vertices (say, to 5 = pentagon), and the type of fill. I’m simply looking for the current UI widget that re-displays the settings that appear when you create the object for the first time (and click the box). These are basic object properties of a mesh-circle object which should be editable after making it.
Oh. OK ! I think I get it. I think it’s possible to disable the display of adjust last operation somehow. It’s enabled by default though. I don’t remember where the setting is and I am away from PC now. If you load factory defaults, it should be on. I think I answered it here https://blender.stackexchange.com/questions/293058/adjust-last-operation-panel-no-longer-appearing Not sure if it’s the same in current version.
F9 key for that
One of Blender’s “dirty, little secrets” is that the Add > Mesh > Circle tool really only creates N-gons.
Reducing an existing edge has a way to remove the edge directly from the editing function.
And there’s Addon to help with this.
I don’t know if there is an Addon available for free.
※ If the generated information is available, you can import the settings using the F9 key.
This is only possible when you have created a Circle and no other functionality is used.
If you want a non-destructive workflow, then install the Non-Destructive Parametric Primitives, “ND”.
They remain editable at any time, no matter what else you have done.
“All right, moving on. I’ve got geometry to create.” Thanks for the input and the patience, everyone!