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Your use of "boiler" is confusing because this can describe either a tank water heater, or a hydronic house heating system which often includes water heat also. But either way...
Time-of-use or variable energy tariff from your utility
Turn on the boiler every day for a small amount of time (e.g. 30 minutes? this is what I am currently doing and I always have hot water available)
And you may notice your phone does the same thing - you get to use it anytime you want, but you only have to charge it occasionally.
"Well of course it does that, Harper, it has a battery!" Well, a tanked water heater (often called "boiler) is a thermal storage load. It has a huge and very well-insulated water tank, and acts like a huge battery - just a hot, wet one.
Now, a great many utilities offer variable-rate tariffs based on time of day, market conditions, etc. So you can set your water heater to do its heating during those times, if it is financially advantageous to do so.
If you are doing home heating with your boiler, you could have a very large and very well-insulated cistern where you keep a large mass of water that you heat when tariffs are favorable. In fact British people have storage radiators, right in rooms, which heat up bricks at night on cheap power, and then radiate that into the room during the day.
Other than that, there's not a huge savings.
And that's because water heater insulation is already pretty good. You're welcome to supplement it with more - there's no fire hazard to adding more insulation to an electric water heater, and they make insulation kits for that purpose.
When we've looked at this question in the past, we just didn't find a better answer than that. The idea of shutting it down while not in use and letting it cool down, then having to reheat it, saves a little... but it doesn't save much. Also it lowers the temperature into the "breeding bacteria" range, which is a health risk.
The best for last: Heat pump boiler. Hands down. Not even a question.
Now if we're talking "water heater" there are complications, because if it's interchanging heat in the utility room it's filling the surrounding space with cold air, which means ???. (party because of free A/C, or run the furnace more, or both depending on season).
However, in the greater space of "boilers" there certainly are outside-air-sourced heat pump boilers, either for water heat or for home heating (though more easily found for the latter, which is typically a hydronic system providing both home heat and water heat). These move the heat exchange outside, eliminating the cold utility room. And yes, better ones can work down to very cold temperatures.