What HK said.
ACCESS is an option, but it only goes through 8th and there's a waiting list. It's in the NE area of town. We don't have direct experience. There's not a high school option (they say there is, but there's not).
Winterhaven is a math and science magnet that accelerates in these subjects. They could be a good option depending on LOG. They are a lottery school, but have a high proportion of identified TAG kids. We have experience here and it's a good school, but not so great if you have a verbally gifted kid. They do about as good a job of differentiation as you'd expect for a class with 35 4th graders.
Metropolitan Learning Center is a very cool K-12 magnet in close-in NW. It's interesting in that it's not a lottery school and you have to apply, but it's public. They are super flexible and seem to skew more 'quirky and creative'. We ended up on the waiting list there when DD was trying for K.
Catlin-Gabel and Oregon Episcopal are both highly regarded, but again not specifically for gifted kids. I know kids who go there and they are getting a world class education (and for the cost, well I wouldn't expect less). But they seem to skew more toward bright and motivated kids rather than highly gifted. They do have need-based scholarships available.
Village Home Learning Center in Beaverton is a homeschool school (brick and mortar classes with teachers but you are a homeschooler). Very different model, but can be a good fit because they are very flexible in terms of age ranges for classes. Not good if there can't be a parent available on a very flexible schedule, especially for younger kids. We have direct and positive experience with this school and it's very affordable. However it's pretty unstructured and you don't benefit from the accreditation. It's also just very different and not every family is going to be able to be that much out of the box.
Another virtual option is Baker Web. I really like this among the online-only options for their creativity and support. We gave it a shot, but found that DD is just too social to not have a classroom. Still a very worthwhile school to look at if you have more of an introvert and someone who can be home.
DD is starting at Pacific Crest Community School (NE) in the fall. They are private and nonsecular, but about half the cost of the big private schools (so only scary, not completely terrifying). They are super flexible and again seem to skew to gifted and quirky/creative rather than high achieving. But they are fully accredited and send their kids to all kinds of universities.
When DD turns 13, we may look at a PCC class. There are a lot of hoops, but it's doable and her new school is the kind of place that would support that kind of dual enrollment.
There are also parochial schools of various types, some of which have great reputations (some of the Catholic ones in particular). Tuition is better, but obviously you have to be comfortable with the religious aspects. We don't have any experience here.
If you can stay out of Portland Public, you may benefit. Lake Oswego has a great reputation (they actually charge tuition for non-district attendees, but it's hard to get in I think). Riverdale is another tiny district in SW. And Beaverton has some really high performing high schools (including an IB school, which may be what HK was mentioning) as well as a program for highly gifted middle schoolers called SUMMA. DD's BFF is attending SUMMA and she's getting a pretty great education. But it's not something you can access unless you live in the district.
Of course you do pay for these schools, it's just in the cost of your house or apartment instead of tuition. Beaverton has the most reasonable housing.
In a nutshell, they don't spend money on gifted education in Oregon and they don't make it easy to research or find good options. Portland Public has some issues, including some rotten options for highschool and a new push to limit in-district transfers (which they are doing for good reasons, but which will only hurt gifted kids in certain schools).
I hope this information is helpful.
