Unfortunately, for bad allergies, I would highly recommend consideration of a HEPA filter, and these are not typically reusable with how they work and how ultrafine particles behave. (There are a few washable filter units available, but the ability of these filters to effectively reduce particulate matter over the course of time, especially following multiple washings, is not really well established and efficiency can decrease. If you did want to go this route, Honeywell offers a few products to consider.)

For really bad allergies and cat dander for a medium room, I would highly recommend consideration of a unit with a very high level of airflow that uses a HEPA filter and has a reusable prefilter, as a prefilter that can catch larger particles and be cleaned will prolong the life of the HEPA filter. The Coway AirMega 400s (or smaller 300s) is a fan favorite, because it uses H13 HEPA filters, has reusable prefilters, draws air from more than one side, moves a tremendous amount of air, is quiet, and is energy efficient. Filter costs are not terrible, and you can get about ~15 months from them, especially if you are vigilant with frequently cleaning the prefilter. Medify, Levoit, Rabbit Air, Winix, and Alen also make some sound offerings, and the Alen BreatheSmart, Levoit 600s, and Medify MA-50 and MA-112 are other stars with high levels of airflow, although I would consider the AirMega to have the most favorable balanced attributes in most cases.

If the price of the AirMega is not favorable, the Winix 5500-2, Coway Mighty 1512, and Levoit 400s are ones to possibly consider. They are not as energy efficient and do not have as much airflow as the AirMega, but are still great choices for medium rooms.

Finally, one budget option to mention is the Corsi-Rosenthal Box. This is a DIY filter made from furnace filters of MERV-13 or higher and a box fan. It will reduce particulate matter, although not as well as a true HEPA unit will, and for your case I would definitely recommend a HEPA unit. That said, it's inexpensive, easy, and would likely still help, so it is worth noting.

I have an IQAir filter. Not at all cheap, but are state of the art, Swiss room size, hospital grade air filters. The outer filter has to be replaced once a year, and the primary I think once every three or four.

However, I would highly recommend allergy drops. These are like allergy shots, but you don't have to go in every day(for a while)/then every week. Instead, they are sublingual (drops under your tongue). They have not solved all my problems, but have solved a *lot*. This is a commitment for at least three years, but helps within a few months, and drops only have to be mixed/picked up once every three months, and are custom mixed for your allergies.

IQAir filter is $1,000, then $100 annually and $400 every four years about.

Allergy drops are about $400/three months, plus initial doc appointments and testing, but may be covered by insurance. If not, at least allergy shots usually are.

The best quality cheap-o solution: OTC Flonase, and a good quality vacuum.

Are you allergic to the cats?

while people seem to be telling you that only disposable filters will be sufficient, i think it's probably worth mentioning that something like an electrostatic particle filter, with permanent plates, or something to that effect, could extend the life of your hepa filter, if used in tandem?

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my rainbow vacuum has a built in air purifier mode, and uses water to filter particles. i'd imagine that something like that would be a great addition.

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radio ionization, iirc, causes particulates to fall, so something like that could be a more permanent addition to your arsenal.

there are permanent electrostatic furnace filters, i guess they are called, that you could be using instead of disposable ones. someone recommended a design with a window fan and a box made out of these square filters.

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just some musings on the subject. i am no expert.

I am in a similar boat. Not what you asked, but I spend for the good house filters and change them every 3 mo. I have air purifiers with HEPA filters in the bedrooms. I vacuum when I have trouble breathing. Also, my cat was urinating on the carpet, and I'm allergic to that. Odoban wiped out the allergy in seconds.

I hope you get the answers you are looking for.

No one can help you as you have fundamental misunderstanding of the industry.

1. Purifiers are sold on square footage of the room(s) used. You didn't list it so 0 people can recommend one.
2. 0 purifiers with a washable/cleanable filter are purifiers. The only standard for purifiers is their ability to filter the air. You can quantify this via the rating, with the 99.9% gold standard being a HEPA or PM2.5 unit which has a filter that needs replaced.

Now if just want a "purifier" that doesn't do anything other than make noise, move air and filter cigarette butt sized objects, you can get you a box fan and tie a washable filter to the back, but make no bones: This ain't a purifier.

Daikin ones perhaps you can try. They claim longer life than others thanks to their streamer device. Not sure if it works. They seem to be using multiple technologies in that device, plasma ions, photocatalyst, etc.

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Interesting question and having quite a bit of experience you need HEPA filter but not only that, it needs to be a fully sealed machine otherwise the dust just leaks out of the machine anyway. I now have 2 Dyson air purifiers based on the fact that they are fully sealed - there is an interesting video showing competitor models leaking air even though they do technically have a Hepa 13 filter.

Alexapure

I have had several Honeywell HEPA purifiers and my oldest has run 24/7 for 6 or 7 years. I think they are pretty sturdy.

May be you can choose the air purifier with a washable pre-filter and clean your room regularly : )