What Cerebus says has a lot of truth and experience to it but it's good to be careful to apply the experience to the particular situation.
For example, if you are going to be a go-getter change-agent the ability to effect change will probably (but not always) be increasingly difficult as the company gets larger. In general, in a company with 10k people it might be difficult for even the CEO to make material change.
Likewise, if you are a whatever-it-takes make-it-up as you go middle manager who is ready to ask for forgiveness more often than permission you might be able to make your approach work better in a company with 500 people than 5k people.
On the other hand, if you are early in your career and you are not exactly sure how the big companies operate (many of which make their size and inertia often work for their employees as well as for the company itself - rather than against the employees) you might have some reasons to work for a big company - 1) you can learn how the big company operates and you can take the good with you later in your career while avoiding or discarding the bad, and 2) once you have the name brand pedigree on your resume with some meaningful accomplishments it will help you land opportunities because the name brand big company pedigree will become part of your pedigree. For every hiring manager who says "golly, I'd never hire someone who successfully performed at Samsung" there will be 10-100 hiring managers who will see it as a positive mark of distinction. In other words, if two candidates early in their career with similar skill sets have 7 years total experience and one candidate has worked at two places no one has ever heard of, and the other candidate has worked at one no name plus Samsung, or Samsung plus Intel, the candidate with Samsung (or Samsung plus Intel) on their resume is going to win all else being equal. (Of course for every “rule” there is an exception: if the hiring manager has come to believe that all employees for all big companies are lazy and incompetent they might take the candidate who they think has shown how to make progress on their own with little or no structure to support them.)
Before going small or flat out entrepreneurial it can be worth getting trained and worth gaining the good house keeping seal of approval from one or more big companies. When you can't learn or earn any more and you have realized that bureaucracy is holding you down, it might be time to go smaller where you can have a bigger impact. Many mid-size companies prefer candidates from large companies because they figure those candidates know where the mid-size company is trying to go. The point is it’s useful to have a longer range view than big is bad and bigger is worse.
Most careers are going to be 40 years plus, so you need to pace yourself. 25 years in with a bunch of 2-3 year tours of duty at places without pedigrees is going to become hard to explain, which won't matter if you don't mind going through the process every 2-3 years. But you might wish you had paced and placed yourself onto a thoughtful trajectory. When hiring managers see a candidate with 1-2 years of experience for the last 5 jobs they are pretty sure how long the candidate will last in a new job.
Each job should ideally last longer than the previous job (if you are getting smarter and picking better) and to make this ladder improvement process last 40 years plus it's good to build on a strong foundation. Don't let cynicism get in the way of a building a coherent career path. To get a job or even an interview someone is going to have to look at your resume and say “this makes sense.” If people have to study your resume to see whether you are growing or going sideways or around in circles they might give it about 15 seconds and if it doesn’t show coherent and consistent growth in terms of your responsibilities it will probably get passed over for the next resume on the stack. If hiring managers have to spend their time looking to understand the business model and the relevancy of the companies you worked for that isn’t going to help either. Of course you can potentially grow your responsibilities and achievements regardless of company size. If you are independent and the best at what you do you are all set as long as there is a sufficient market for what you do. But if you need a job with a company you are going to have to face hiring managers and the hiring process. Hiring managers can determine that a sergeant in a big army might have held more responsibility in a captain in a smaller army, or vice versa. The key is not what size army you worked in; rather you should be looking to make progress each step of the way in terms of your responsibilities and achievements, otherwise someone might look at your resume and say “peter principle.” When that happens you might not be on the ground floor but you might have hit a ceiling. When you are young the concept of a ceiling can be pretty abstract, but that doesn’t mean it can’t occur.
As long as you are learning, earning, and having fun doing something you believe in you should be in pretty good shape – but the trick is keep in mind that life is a journey not a destination – and the people who will decide to keep giving you tickets for more travel will increasingly look at where you have been and what you have achieved to determine whether they are going to let you go on to their next station - so it’s good to show some coherent growth. Point being, think “career” as much or more than “job”, and have a sense of how your career can potentially keep advancing. People can recover from incoherent career growth but why set yourself up for more hurdles if you can avoid them by planning ahead. As Yogi Berra (probably more popular in the U.S. than Russia) said “unless you know where you are going you could wind up somewhere else”.
When you are ready to go full speed entrepreneur including building your own company and raising money the experience, knowledge, contacts, and pedigree gained from name brand companies might also help you gain the confidence of investors. Or alternatively the investors might say “well, he can be the CEO for now but we will have to replace him with someone who has experience in an environment with scale.” Ideally but not always you will know this thinking when the investment is made. Of course if you worked for small companies and won business from big companies that’s another way to earn big company pedigrees.
In the end, there is no one route for everyone but not all big companies are bad for all careers and early in a career they can be outstanding for multiple reasons, not the least of which are learning, earning, and the development of your pedigree.