I am sorry to hear that Romeo ingested some Aleve.
Aleve contains naporoxen, which is a nonsteroidal anti-inflammatory that we do not use in dogs or cats due to the high incidence of side effects including gastrointestinal bleeding, ulcers (sometimes perforating) and kidney or liver damage even when given appropriate doses.
The dose listed in Plumb's Veterinary Drug handbook in 2mg per kilogram (2.2 pounds) of body weight every 48 hours.
We can see toxicity symptoms of stomach ulcers at doses as small as 2.5mg per pound of body weight, and kidney failure at doses as low as 7mgs per pound of body weight. Death can occur at doses as low as 75mgs per pound of body weight.
Peak blood levels are within 3 hours of ingestion so these levels should have occurred the first afternoon and would have explained his continued vomiting after that first dose.
Toxic side effects are abdominal pain, vomiting, gastrointestinal ulcers which can perforate, dark tarry stools, weakness, anemia, and kidney damage shown by increased kidney waste product levels (BUN and creatinine), and liver damage shown by increased liver enzyme levels on blood testing. Cats seem more likely to have kidney damage rather than liver damage, but both can occur.
Unfortunately the half life (the amount of drug metabolized and decreased by half) in cats is very long, at least 72 hours, and as such the effects of the drug linger.
At this point he is definitely showing signs of toxicity, he needs to see his veterinarian for bloodwork to evaluate the damage done and supportive care.
Your veterinarian will examine him, look for blood in his stools and urine, test his kidney & liver waste blood levels and urine for signs of kidney or liver damage and get him started on sucralfate which is an ulcer coating drug, as well as an acid reducer such as famotidine or omeprazole. They may want to put him on intravenous fluid therapy to flush out any remaining drug and support his kidney function. If he is continuing to vomit they can give him injectable anti-nausea drugs.
They will also likely recommend feeding a very bland diet.
At home now you can start either:
1) Pepcid-ac (famotidine) at one quarter of a 10mg tablet per 5 to 10 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at one quarter of a 10mg tablet per 5 to 10 pounds of body weight every 24 hours.
These are very safe and should be given for several weeks to protect his stomach as it heals.
But he really does need to his veterinarian.
Best of luck with your fellow, let me know if you have any further questions.