They dim. They stay the same.
Work Step by Step
For two or more devices connected in series, the supply voltage equals the SUM of the voltage drops across the devices. Each identical bulb "feels" a smaller and smaller fraction of the battery voltage. They become dimmer as more bulbs are added. In other words, adding more bulbs in series means the resistance goes up, the current goes down, and each bulb becomes dimmer. For two or more devices connected in parallel, the voltage is the same across each device. This is Rule 1 for parallel circuits, on page 444, and discussed on pages 443-444. This is also illustrated in Figure 23.18. Each bulb still feels the full battery voltage, even as more bulbs are added in parallel.