Time travel requires the creation of a closed timelike curve (CTC): a closed loop in which spacetime returns to a starting point.
Let's assume that a CTC is possible (an open question in physics), and it allows a single signal (yes=signal, no=no message) to be sent to a time machine that's already operating, after which the connection ends. What can be done with this low bandwidth connection?
Say your goal is to win the lottery. You start your chrono-receiver and buy a lottery ticket. If you win, you send a yes. If you don't get a signal, you increment the lottery number and try again. Even if your chance of winning is 1 in a billion, you will eventually get to the correct number.
What do you do after winning the lottery? Solve the next problem. Anything that is physically possible for you to becomes trivial given enough attempts. You just need to build an Ideal Solution Database (ISD) to keep track of successes and ensure each attempt is unique. You are using the CTC to perform computation on an infinitely powerful computer. You can't go back to before the time machine was invented (that's how a CTC works), but you can optimize your action post-creation to achieve any and all outcomes that are physically possible. (If this is confusing, watch the Rick and Morty episode "Edge of Tomorty: Rick Die Rickpeat")
Where does the energy for this process come from? Where does the entropy go? From a thermodynamic perspective, time travel is problematic, whether you want one trip to make sure your parents have that first date, or a quintillion trips to become God-Emperor of Earth.
So this is the basic problem with all time travel in fiction. Even if you don't set out to create the ISD, the temptation of optimizing any action leads toward the creation of an ISD. As each goal is achieved, the next goal is brought forward in time. The only limitation is the time to record each goal in the ISD, and the process can be used to optimize the ISD too. History compresses into a singularity and the flow of time as we know it ends.
While the time loops may be infinite, the ISD calculations and actions still generate entropy, so the ISD civilization has an expiration date. Assuming the civilization remains in a particular area of space (such as a solar system), it will perform all work possible until reaching heat death. From the perspective of an outside observer, the ISD civilization accelerates to a singularity, then vanishes.
If a CTC is impossible, what's the point of this speculation? All intelligence tries to approximate a CTC+ISD.
When you try to throw a basketball into a hoop, you first create a simple model universe in your mind that simulates the trajectory of the ball, then test the hypothesis by shooting the ball. You repeat the process, using simulation and testing to perfect the ISD in your mind. All intelligence works by running simulations, testing them, then creating a solutions database from the results. Unlike a CTC, each iteration has takes time and uses energy. To minimize this cost of simulation, civilizations are likely to trend to ever more efficient computing, bound only by Landauer limit, the theoretical lower bound on energy consumption.
Currently, we note a big difference between simulations (whether in our mind, computer software, or a physical system, such as a wind tunnel) and reality. However, a future civilization which exist entirely as software, and may convert the fabric of reality into computational substrate (aka comptutronium) may not recognize such a distinction. If the Landauer limit is somehow overcome, future civilizations will achieve what is effectively a CTC+ISD.