I would think portions of this are somewhat achievable via Arbos's Rewards addon already. Perhaps that should be the starting ground for the rest of it?

Could well be. I am filing this on here as a potential project brought it up, and while it's a fairly standard thing in gaming (and hence not confidential), it's not common for CMSs. I think it's a really interesting feature so it's worth filing whether this project happens or not.

A block for showing the latest achievements awarded to people.

It may be far more common for gaming but I see several uses for it on other site. People crave recognition and this is a good way to give it and can help to facilitate community interactivity. Non gaming applications I had planned for something like this:
On a recipe site of mine: Awards for most (or certain number of) recipes submitted. Same for certain number (or most) recipes voted best and a few other things like that.
Here' on Composr you could give badges out to those in the community who have submitted themes or addons or reached a certain point status as helpful soul etc.
I think this is an excellent project idea and I intended at some point in the future to work on Arbo's decent beginning for improvement. But right now I have to get a couple sites up and running and finish this tracker port.

Note that your points above overlap with what the gifts addon does. I haven't actually looked at Arbo's badges addon properly myself.
Also in Composr you can assign awards to a member, which overlaps slightly to.

Awards: Manually assigned by admin to an item of a particular content type, happens to be applicable to members
Gifts: Members may buy each other gifts in the point store
Achievements: Automatically assigned badges for members
Rewards: (I'm not sure, I've not looked at it in detail - not to take from Arbo's good work at all, just I am usually too busy to look at contributions, I wish I wasn't)

It kinds of irks me having too many ways of doing the same thing, somehow these can probably be brought together. Gifts is probably separate, but perhaps rewards and achievements would best be implemented as auto-assigned Composr awards (i.e. on top of awards).

I agree it would make more sense with less interfaces to the same types of operations. I haven't played with Composrs "Awards" system enough yet to understand all it does but I basically thought it was more of a "Featured Content" type thing that just makes the content prominent for a set time. I wasn't sure if it left any trail in the users profile as a sort of badge of recognition or not?
I do know that the "gifts points" thing sot of works a little like it too but it is still not quite the same thing either. And at least it is not as pretty has having a profile page with Cool designed "Badges", "Medals" or "Trophies" etc that truly stand out as "Achievements".
If no-one tackles this before I have the time to look at it I will certainly look to see how all the separate parts could possibly work better together for sure.

Adding one more note that I thought of sort of falls into this recognition system of rewards is probably member titles. Maybe that also needs to be looked at somehow integrating with these (Let's maybe call them) "Recognition Ideas" in their interfaces somewhere?

I'd like to see an extension of the usergroup-upgrade system used with this.

i.e. members get their user group upgraded (and therefore can access different things) once they've unlocked certain achievements.

"I wasn't sure if it left any trail in the users profile as a sort of badge of recognition or not?" under the trophy icon next to the title. It'd probably want to be put on a profile tab, but that could be done while keeping it award-based.
Awards can work both as featured-content (the latest content given an award), but there's also a browsable archive, and it also shows under these trophy icons.

"Adding one more note that I thought of sort of falls into this recognition system of rewards is probably member titles."
Hmm, I guess you could also have it so an achievement optionally auto-changes the title.

"i.e. members get their user group upgraded (and therefore can access different things) once they've unlocked certain achievements."
Probably the best way to do that would be to make a usergroup promotion target depend on both the point figure needing to be reached, and also a selected set of achievements to have been reached.

On one CMS I had been working on a modification that does something like this (unfortunately the code written for it would not be portable to this as it was nuke based) But there "Ranks" (essentially user titles here) were separate from groups and the mod I was doing was part of a rewards system (though I called awards) and it integrated with other mods like "Jobs" (or duties as I called them) which is a little like a member title/group thing but anyway with jobs you could assign a user a monthly allowance (as well as one time entry bonus) of points when they were assigned a job/duty. Now that I think of it the Ranks was the part I was working on and Ranks could be manually set or Auto set based on criteria (days a member, post made, points earned etc etc) A rank could be assigned a promotes to next rank automatically upon reaching the achievement set for that rank. (note achievements could be chained ie you reached 500 points and were a member for at least 3 months type thing. A rank had a default user group set to it and an upon promotion remove from old group setting option (or if unchecked would leave it as a secondary) it could also have a default duty/job attached to it for the allowance thing. Also a badge (or award) could be awarded upon reaching the rank (I mean one that exists outside of the rank title and image). I think this is quite like what we are talking about here in many ways.
The interfaces to each were closely tied.
You added set up you awards (medals ribbons badges etc) under one. The just basically involved title images descriptions (I think no automatic criteria management).
You set-up duties/jobs again title description and a wage in site points offered earned while doing job
Ranks is where the tie ins all cam. when you made a rank beside title desc, image you defined whether it was an auto rank or manual and in which order it reside in that tree. Upon promotion to it Which Groups (if any) would you be automatically added to) and which duties. (what bonus in points were you allowed to receive? and if set to auto based on what criteria (points < or > or = something (posts < > = to, Time a member < > = to etc.)

I really liked that system although the coding for it had only reached about 80% completion before I had to drop the project for medical reasons for a long time.

So anyway sort of following along along with what is being discussed here I think a good approach maybe:

If needed modify Arbo's Rewards addon but use it as the base for defining the Awards (Recognition Awards I mean like badges tropies, ribbons, whatever have you that shows up as a separate profile tab (as it does now).
Modify the Composr AWARDS system that when something is awarded it ties to an award/reward badge from the arbo rewards to get added to users profile
Modify member group promotion to allow for earning badge on arbos addon and or maybe vice versa (if earned a reward be added to group?)
Member title - Again add Reward or Reward adds title?

I know this is what some of you have said just recapping.

Actually once we iron out a good probable direction maybe I'll go back and edit the final note with the list of ideas and delete the rest to save this from becoming too long to scroll lol

By all means move on this if you want to Duck, you have more time to think of these kinds of details than I do :D.

I can't promise I can do anything about for at least the next 3 or 4 weeks but if in that time I can I will.

(Unrelated to the above discussion)

It may be an achievement is just a gift that the system gives automatically. Therefore, this may all be an extension of the gifts system.

It might also be that gifts are worth a certain number of points. Therefore it costs that many to give them, and the recipient receives those points along with the gift being displayed.

I know this item is growing a little stale in the tracker and I offered work on it (and a couple others I know) but I just have not had time and won't for some time more to do much. I would like to get started on some of them but until I can get other projects complete I can't commit to anything else. I am sorry to leave items in the tracker long but if no one does get to these types of ones I do hope to come back and tackle them at some point.

I'm closing this, as it's getting reconsidered as an extension to #871.

I find admin gamification more exciting (as getting webmasters more engaged is very important). And if that's done, it can then be extended to a more general purpose achievements system.

Largely the ideas remain the same, just it would be a different route toward implementation.