What you are describing is an imbalanced power situation, where one station (the access point) is transmitting at a higher power setting than the others. Increasing the transmit power of the AP will not really increase your coverage as your client has to be able to transmit back to the AP as you indicated.
However, if the majority of your traffic is downstream (from AP to client), then this can sometimes increase performance. The reason this can help is that the larger data frames (in this case the majority of traffic) can be transmitted and received more reliably at higher data rates, with less retries/loss.
Be careful though as this can also lead to a number of situations where you will actually hurt your performance. For instance, setting the transmit power too high (higher than the manufacturer intended) can increase the distortion of a signal (especially with cheaper hardware). In addition, some hardware will not allow you to increase the transmit power above a certain point, even if the firmware seems to indicate you can.
Generally, you are better off using a passive means of producing gain, such as a higher dBi antenna. This will produce an increase for both transmit and receive signal strength. Make sure you understand how this will affect your environment (for instance a higher gain dipole will get you more signal strength on the horizontal/azimuth plane than a lower gain dipole, but it reduce the vertical/elevation providing less coverage above/below the plane of the antenna).
As for a repeater being a must to solve weak signal problems, I have found that often repeaters create more problems than they solve. They will typically decrease performance of a busy wireless network and can introduce other problems as well (hidden node, etc).
Instead of repeaters, I would recommend additional APs to provide additional coverage. You can do this by planning the placement of the additional APs or using directional/semi-directional antennas forming an "AP array" from a central location.