Almost-huge cardinals are characterizable in terms of coherent towers of supercompactness measures, with a certain property of the direct limit model (see Kanamori's book). A useful large cardinal hypothesis between almost-huge and huge is the assertion that a cardinal has an almost-hugeness towers of unbounded height. The most natural name for this would be "super-almost-huge," but that sounds too awkward to me. I was thinking "ultracompact" might be good because of its affinity with supercompactness. So my question is, is this a bad choice of terminology for some reason, or did someone already give a name to this hypothesis?



Is "ultracompact" taken?
I think that the super-almost-huge terminology is fairly widely used with this meaning, so that $\kappa$ is super-almost-huge if for every $\theta$ there is an almost hugeness embedding $j:V\to M$ with critical point $\kappa$ and $j(\kappa)\gt\theta$. 


This terminology aligns with the similar terminology for [superhuge](http://cantorsattic.info/Huge#Superhuge) cardinals and [super $n$-huge](http://cantorsattic.info/Huge#Super_n_huge) cardinals, which seems to be fairly established. In addition, there are the [super high-jump](http://boolesrings.org/perlmutter/files/2013/04/ChartForWebSite.jpg) cardinals, and [super-almost-high-jump](http://boolesrings.org/perlmutter/files/2013/04/ChartForWebSite.jpg) cardinals, which follow the same idea of having these embeddings for arbitrarily large targets. For this reason I would recommend against introducing a new terminology without very good reason.


(Meanwhile, note that "hypercompact" is already taken, as you can see on Norman Perlmutter's chart of large cardinals near the high-jump cardinals (linked above), although to my knowledge "ultracompact" is still available as a large cardinal name.)


