What's a good way to ask for work when you completed your task without sounding repetitive and pushy

Having had a similar issue in the past, what I have found works well is creating a backlog of prioritized work so that both you and your manager know what you are working on, when you are finished, and what your next task is.

With something like this in place, you don't need to interrupt them and force them to come up with a task for you, everyone already knows what is next. When you only have 3-4 items left, you can say something like "Hi _, my priority list is down to just a few items, when you have a moment tomorrow, can we meet on the next items of priority, or can you add to it?" This way it is at their convenience to add to your list.

Your boss may not want to work this way. To consider it, you may wish to start a conversation like "I noticed I interrupt you often when my work is completed. Would it be better for us to set up a list of items that I can pull from so I am not interrupting your workflow?" They may say no. In that case, you can simply ask in this discussion if they have a preference for how you get your next task. They may prefer things as they are.

I usually message my client manager on Slack saying, “I’m done with my tasks for the day. Please let me know if I can work on something else.” However, I worry that sending this message frequently—sometimes daily—might come across as repetitive, pushy, or inconsiderate of his busy schedule, especially given the cultural differences and his workload.

So instead, ask something like "In the future, if I have completed my tasks for the day, what would you like me to do?"

That gives your client manager the opportunity to suggest an approach that doesn't bother him.

He might want you to check in with him each day. Or he might come up with a checklist of future tasks that don't require checking in daily. Either way, it's his choice and can't be inconsiderate.

You are totally overthinking this. I would wager, with certainty, that you have to get your timesheet approved each week by this same manager. In the old days (per modern standards) this would have been a piece of paper. Electronically now, it could be an email with an attachment (still crude-ish) or some web page that the manager goes to in order to approve the hours. You do this week by week. It's repetitive (I am using your word, on purpose). You have no qualms about doing this because if you don't, you don't make any money.

Just the same, a reminder to this person to keep you engaged in work also keeps your rent (or mortgage) and bills paid. You're not shy about the timesheet, so why be shy about the reminder? Once or twice a week will do. When you run out, let him know. Wait two days, and remind him again. You have a good point that a daily reminder would be too much, but I don't think anyone would fault you if you keep the reminders at no more than twice a week.

In the bigger picture, you will have situations where the manager takes time off and leaves someone with authority to sign your timesheet, and you run out of stuff to do. This is guaranteed to happen at some point, and is just normal course of business. You can't be in control of everything, and all you can really do in these cases is to make sure you're putting forth your best effort for the things that are assigned to you. That's it. The rest -- let that go! Worry doesn't solve anything.

Good luck.

Postscript:

Another commenter has suggested getting a backlog of things to do from the manager. This is a great idea in practice, but doesn't really work in reality because the manager can give you a list on one day, and then the business priorities can change on the next day -- which would mean you're not doing the work in the order of actual priorities. With such a list, you lose the communication loop, OR the manager has to constantly adjust the priorities on your list each time they change (which is time consuming). Again -- it's a great idea in theory, but it's not realistic.

If this situation has happened more than a few times and is likely to happen in the future, then talk to your manager and ask how you should be filling this time to complete your billable hours per day.

It's entirely possible that you may be allowed that time for yourself (especially if your unpaid overtime is visible), or you might be asked to only bill the hours you actually work. Or there might be other planning/preparation work you could be getting on with without the need to pester for work items on a nearly daily basis.

No matter what you work with - whenever you are working in project form, there will always be situations when the project is coming to a halt. Maybe you are waiting for customer feedback, or for a colleague to finish something, or for tech support or for some delivery, and so on. And sometimes like when you are stuck with a problem, you just need to take a pause from the project and go do something else for a while.

Any half-decent manager ought to be aware of this - so if they only hand out one single task at a time, they are doing it wrong. You should ask them if there is lower priority work you could be doing when your main project is coming to a halt or when you are between projects. You can probably propose a few things yourself - it looks better if you ask "when I have some time over, is it ok if I work with x" rather than "I'm done with my task". Employees taking initiatives ought to be well-received.

When I worked as manager for an engineering team, I always made sure we had a couple of lower priority projects in the background like that, for the employees to work with at their own discretion. Often these would be related to improving internal routines or improving/evaluating existing/new tools etc, or the ever-present backlog of lacking technical documentation - stuff that really was quite important but not urgent.

Eventually the team members might start to come up with ideas of their own for how to improve whatever it is you are working with. If someone comes up with an idea for improvement and gets a green light from the manager to go ahead and work with it when there's time over, then they are usually working quite enthusiastically with that idea and therefore produce good results - everyone wins. The manager just has to monitor that they aren't getting too carried away, so that working on the spare project comes at the expense of the top priority project.

It is very bad to say, that you are done. That has a flavor of saying, you do not want to work.

Instead, what you should do:

"Hey the task X is done, however that X' might need some polish to work more robustly, although it is not an urgent matter. I am on X', but if there is more important, I would do also that"

If he says nothing or accepts X', then you will "work on" X' on the rest of the day.