To me, as a designer, balance means a couple different things. In a single player game, it means that the player has interesting and valid choices available to them. For multi-player, it means that given equal skill, from the beginning of the game, each player has an equal chance to win. Its a bit more complicated than that however, so let's dive in.
Single player balance is very tricky to approach, especially compared to multi-player. In multi-player, you want to make sure that each player has an equal opportunity to win (usually) and so you tweak values and numbers so that you get the desired result. With a single player game, you generally want them to win.
You can make them godlike in their power, and you've achieved your goal of making them win. However, it is less likely to be an interesting game for the player, as whatever they do, they win. This means that straight up giving your players infinite power generally isn't ideal. However, letting them unlock that power is often ok, particularly when they have to work for it. In Saints Row, by completing most of the content in the game, and by saving up, you can unlock infinite ammo, instant reloads and not take any damage. This power trivializes the rest of the game (if you haven't already finished).
In Mass Effect, you can modify your guns so that you never have to pause your firing. This power doesn't trivialize the game, but instead rewards the player for making a particular choice and seeing it through. Similarly, in Skyrim, if a player chooses to focus their efforts into sneak attacks and one handed weapons, they can do over sixty times the base damage of their daggers, letting them do much more damage than nearly any other source in the game. On the flip side, if they aren't able to approach a combat situation as they would normally choose, they are at a very large disadvantage. While the player has the ability to instantly kill a dragon or giant, it is dependent on their skill.
Additionally, in Skyrim, if the player does not choose to go down the route of sneaking daggers into the backsides of their enemies, they can still defeat the dragons and giants. Being a wizard is effective in a different way, but still works well, as is fighting with a one-handed weapon and shield. Choosing which route to take is a choice effects the way the game is played (making it an interesting choice) while not restricting the player's ability to progress (making it a valid choice).
When playing Minecraft, you can choose to turn on Creative mode, which instantly gives you infinite access to all the resources in the game in addition to giving you flying and making you immortal. However, because the game has two different goals, this power doesn't completely trivialize the game. For some, it makes it more interesting. If a player isn't particularly excited about adventure, exploration and taking down the Ender Dragon and Wither and they would rather build glorious structures that take days of real time to build, having access to the resources enhances the game, rather than taking away from it.
In Saints Row, after unlocking the power, the game becomes much less interesting, as you don't have to concern yourself with the mechanics at all, you just enter a fight and walk out victorious. With the advantages earned in Mass Effect and Skyrim, while your character can be very powerful in particular aspects, it comes at a cost of other avenues of power. With Minecraft, the toggle can ruin one aspect of the game, while making another much better.
This post is getting to a decent length, and I haven't even started in on multi-player balance, so I'm going to call it here and get another one going on further aspects of keeping things even in the near future.
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