I would like to know the CHEAPEST way to go from Amsterdam to Paris. But it don't need to be straight, only cheap.
For example, I have no problem to make a tour from Amsterdam until reach Paris.
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The cheapest way to travel from Amsterdam to Paris is by hitch-hiking.
Failing that, early-booked buses are cheapest. For example, if I search now for a Eurolines bus on a weekday late May, I find tickets as cheap as 9€. You want to search for Promo tickets, indicated in dark red in the fare calendar.
Screen-capture of Eurolines booking system
There are probably other bus companies, but probably it won't get much cheaper than that.
This answer is worked out from several comments on the question, (which are often found in any request to get somewhere cheap.)
The distance Amsterdam to Paris is roughly 500 km, by the main roads. But as you will want to good route to cycle, you may add a few km, so say it will be about 600 km, about 375 miles.
For those who are experienced cyclists and know how much they can do on a day, it will be easy to work out how long it will take you. It is relatively flat, getting less so when you go farther south.
Google maps gives a height indication of 140 meter, 460 feet. You will get down to almost see level at times, but it is nowhere near the heights you will have to do in high hills or mountains.
One thing you will have realize, there will be winds, often blowing from the south and they can be strong any time of the year.
For those who are less experienced, by doing 60 km day distances, it will take 10 days to get to Paris, and that is a distance almost everybody with some cycling experience can handle, real new cyclists may have to go further down on the distances. On a fitting bike with the wind from behind, days of 100 km are well possible.
If you do your cycling outside the main holiday season, you do not have to make reservations and you can adjust your daily distances to the way you feel. And you will be near rail lines where you can take the train with your bike if it turns out you do not like the experience.
Now about costs. If you happen to own a bicycle located in the Netherlands, it will only cost the overnights and food you would usually spend money on while on holiday.
If you do not happen to have such a bike but have a few weeks (and/or local friends) to find a second hand bike, to your requirements, you can often find one for relatively little money. (I have seen samples from about €25 for bikes that are good enough to undertake such a journey, but you will need time to find such an offer, do not expect to find a good cheap bike in a day or two.)
Renting a bike is also an option if you intent to return to Amsterdam, it is possible to return by train, without reservations, but you will have to change trains a few times. (I do not think the fast trains take many, if any, bikes yet.)
As far as I know there are no 'one way rentals' for this distance yet. There are organized tours which will transport you (and your bike) back to the start, but those will be more expensive and less free to select your own route. Some of those tours will supply you with a bike to use, it will not be the best bike you ever used but it might appeal to some people.
There are many long distance bike routes, this site has many of the Europe wide ones as well as links to national organizations which will have more routes still. Searching on 'long distance bike routes' will find you many more (as well as the organized hotel and cycling tours.)
There are many long distance routes which, often in a combination of several, will cover the Amsterdam to Paris.
It still is about 500/600 km, 310 to 375 miles, or as much more as the more less direct but nicer to walk paths may take. This is quite a lot for most people to do in a short period of time. But again, if you are experienced, you will know your daily distance and the level of elevation will still be the same as for cyclists, 140 meter or about 460 feet.
Get a good guide on the long distance route(s) you intent to use, so you can find information on places to sleep near your route. (Hotel finding sites will work, but you may need to find places which are not on those sites, in case you are in an area where hotels are scarce.)
Like for the long distance bike routes, there are websites for long distance walking routes. Wikipedia has a few for the international and national long distance ones in the target area (and well beyond.) Again a search, like on Long distance walking and a country or route will get you more information. The long distance walking routes on those Wikipedia pages often have booklets for the route and additional information. But more and more the information is found online as well or instead.
Unlike bus (or any public transport) traveling by bicycle or on your own feet does allow you to see the country you are traveling through, visiting the places you stay overnight, maybe even have a stop or two in touristic places, if you are willing to leave your (bike and) gear unattended or stay near enough to make it a walking or bike ride without gear.
Camping will work, both for cycling and for walking, but it will increase the load you need to transport and the preparations and (air) luggage you have to bring. Hotels, hostels, B&B's, airbnb, and even warm showers (places to stay for people cycling) and alike for walking people, will be easier, as you mostly just need your sleeping bag.
Many people will combine walking the better parts of the routes with public transport for the rest, which will likely be a lot more expensive than the cheapest long distance routes for the whole Amsterdam/Paris distance.
While this is a cheap way of travel, people will often spend more on places to stay and food, and it will certainly take a lot more time, several days or even several weeks when hiking and not doing extreme distances each day. It is actually not a way to get to places but a way to spend your time and get somewhere almost as an afterthought.