Should I bring prescription drugs with me for a long trip?

Hong Kong customs states :

Pharmaceutical products and medicines imported in the personal baggage of a person entering Hong Kong and which are accompanied by him and in a reasonable quantity for his personal use may be exempted from licensing requirement.

You take 3 pills a day, and you have a long trip, it is perfectly expectable that you will not have issues having a 2-month stock on arrival. Although this is and will still be at the mercy of the customs officer that will process you

Please, make sure that your medicines aren’t in the Dangerous Drugs list

If they are, you will need a permit :

Importation of dangerous drugs requires an import licence and import certificate issued by the Department of Health. A valid export authorization or diversion certificate from the exporting country will also be required.

Japan’s

You can bring your personal medicines into Japan once certain
procedures are completed. This is because there are patients who need
continuous medication and travelers who need medicines as belongings,
however, when you bring narcotics and psychotropic described in 4,
please note that importing them from overseas is subject to control by
other laws and rules.

To be more specific, when you bring your personal medicines into
Japan, you have to apply for so called "Yunyu Kakunin-sho", a kind of
import certificate, and receive it before you leave home, and declare
it to the officer at customs.

When you bring your personal medicines into Japan up to the amount
described below, you do not have to apply for "Yunyu Kakunin-sho",
instead the officers at customs check the amount of your bringing
medicines, and then you can bring them into Japan.

• Poison, Deadly poison, Prescription drug : Up to 1 month supply
• Drug for external use (excluding Poison, Deadly poison, Prescription drug) : Up to 24 per one item
• Injectable drug and Syringe for the drug: Up to 1 month supply * Only "Pre-filled Syringe" or "Self-injection Kit"
• Other Drugs, Quasi drugs : Up to 2 month supply

In addition, no product containing the designated pharmaceutical
ingredients (see linked table)(57KB), which is sold in foreign
countries/regions with advertisement on enhancing cerebral function
and other mental effects, shall be imported, unless use of such a
product is complying with doctor’s prescription or instruction.
However, it would be permissible for foreign travelers to bring such a
medicinal product into Japan by his/her self, in the purpose of
self-medication during his/her stay.

Regardless of the description above, you cannot bring prohibited drugs
and controlled drugs into Japan.

Regarding details about application for "Yunyu Kakunin-sho", please
check "Q&A for those who are bringing medicines into Japan", which
includes application forms.

"Q&A for those who are bringing medicines into Japan": WORD(149KB) PDF(529KB)

For Japan, you will have less than 2 months worth of supply, which
will not require a permit

EDIT: The phrase above is totally false (thanks @jcaron), you will need a "Yunyu Kakunin-sho" for your medication because you will have a stock of above a month of prescription drugs at the time you enter Japan, everything in explained in the linked site

Make sure to have your prescription ready and if possible in English and the permit, especially in case your medicine is mentioned in this paragraph and as usual check for controlled/prohibited substances

In addition, no product containing the designated pharmaceutical
ingredients (see linked table)(57KB), which is sold in foreign
countries/regions with advertisement on enhancing cerebral function
and other mental effects, shall be imported, unless use of such a
product is complying with doctor’s prescription or instruction.
However, it would be permissible for foreign travelers to bring such a
medicinal product into Japan by his/her self, in the purpose of
self-medication during his/her stay.

The Thai embassy in DC states :

You will have way less than a 30-day supply, provided you don’t need or have applied for the respective permit, you are fine, after a check on the mentioned website of the legality of the drug

Going by the official language, you will have no foreseeable issues

Make sure to have the original prescription and if possible an English translation for it, that may help, and also.

As a general rule, never carry pills without being in a clearly labled container (usually the medicine box), this is the recipe for being treated for a drug-dealer

This is extremely well explained here

If you have still some medication on the return flight to France, make sure to also check the import rule of France, which are here