The reward for doing good work and delivering results in your current job is your paycheck, not a promotion. A promotion is given when you have shown that you have mastered your current role and can step into a different role. Would you say that you have never been good at, or liked, "playing the game" to get promoted? Chances are you have been playing the game, it just levels up and you need to adapt to the new challenges. Here are a few tips to that might help...
If you don't like games, give a different label to the necessary package of skills and qualities required for a promotion. By labeling behavior as a game, and one that you won't play, you have decided to lose. In dismissing behavior as a game, you might be missing important clues as to what would be required of the next higher position. If you think you want to be promoted, you need to work around roadblocks, especially the ones in your head. If you don't play games, come up with a label that will position you for success in keeping with your values and taste. So, if you like puzzles call it the "success maze" or if you like cooking, call it the "recipe for success." Replace "game" with a label you can stomach - and move on.
It's safe to assume that what is required in the next role is not what was required to land your current role. As Marshall Goldsmith so famously said "What Got You Here Won't Get You There." What are the requirements for a promotion? What do you need to be able to do in the next role? You need to understand what work you will need to stop doing and what new work you will start doing when you get that promotion -- and you need to train for that now! Perhaps you need to schedule more time with the boss to study and research what is required in more senior roles. Ask for advice, and stay tuned into what the more senior ranks are focused on. A bonus that comes with speaking to your managers more is that they will get to know you better and recognize your drive and initiative. Once you know what lies ahead, you can decide if you want to pursue a promotion. If what is required is something that goes against your values or code of ethics, then by all means excuse yourself from the competition with your self-esteem in tact.
If you want to pursue a promotion, you need a plan to simultaneously do a great job in your current role while learning about the next role. As you are promoted, you will be expected to prepare your successors and prepare yourself for leadership. Leadership positions demand a more public role - keeping your head down and getting the job done may not be enough at higher levels of leadership. You may need to lift your head and carve out more time to look up and outward to the future. If you are struggling to make this happen, start small. Carve out fifteen minutes a day to study for that next role, to chat with a manager, to reach out to someone in your wider industry network, or to research what is going on in your industry. Choose one aspect of the next role to learn and practice that. Find a mentor at work or in your industry. Consider hiring a coach to help you with your development.
Whatever way you find palatable to continually learn and develop new skills, that's all that really matters. Don't like games? No problem.