I don’t know if we actually had alien overlords or only a small infestation of aliens, but they were all over the place where I was: a large complex, like a conference centre, large enough that a little train ran to some faraway parts. Different kinds of aliens: one we talked to looked most like the head of a green-and-yellow cartoon frog partially submerged in a swamp.
At night the complex was flooded with some kind of gas the aliens could breathe, though during the day they seemed to be able to breathe the same air as us. A man who had been living there for a while took me and Spouse on the little train to the other side, where there was a sealed place for humans to spend the night or to live permanently if they liked, called DEK (in capitals, perhaps an abbreviation but everybody pronounced it as a word). He sang the praise of DEK the whole train journey long, so I wasn’t surprised (it was that kind of praise) when we were herded in by uniformed people with blasters and made to sit down on ugly orange plastic benches around artificial trees with yellow-and-black notices in four languages (English, Dutch, something else in Latin script, and something not in Latin script) nailed to them, detailing the Rules. They started “DEK is a common room” but degraded into fairly harsh rules quite soon.
One of the rules was that you weren’t supposed to carry more than five objects into DEK, and the man warned us that if we appeared to search our pockets or count things while the guards were looking they’d strip-search us and confiscate everything, even if it was fewer than five objects. We were both wearing our summer coats, so we took this to heart, though I knew my pockets were almost empty and Spouse had only his wallet. I don’t know if I had my handbag, or for how many objects it would count.
Later, either in Part II or in a different dream, I was in a place with narrow dark corridors blocked by walls of stacked water bottles that dissolved (discharging the water) just before one walked through. Each bottle wall was a different brand, or at least a different style of bottle. At the end of the corridor that I traversed several times there was a side passage containing a huge bottle of water, at least two and a half meters high, that didn’t disappear.
There was a cute dog in a dog daycare, called Nora (or perhaps Olga), who we (not Spouse by that time, but Prima and Prima’s new boyfriend and me, and possibly one or both of Secunda and Tertia) visited: the dog came to the window to doggy-smile at us and the daycare keepers, a large middle-aged woman and a smaller young man, indulged that.
There was a plexiglass thing that was probably a trick bottle which Spouse did tricks with, taking off a part that looked one shape on the bottle but was a different shape on its own. It was also a puzzle: turning parts of it made ball bearings fall through holes (and eventually out at the bottom). This was supposed to be very difficult but I found it easy, even when I was later handed the same bottle completely covered in intricate beadwork made of letter and number beads which worked as keys to type in secret codes.
There was something that seemed hugely important at the time, so much that I came back to it several times after waking up briefly, but it seems a small thing now: throwing different coloured powder into a corner of a space where it coalesced into small fish-shaped blobs of that colour, which would run together and disappear when they passed a certain critical number. It might have been intermixed with one or more of the other parts of the dream(s).
 Not “power” as I had before editing.