Initially, the only such device was a diode with only a filament (cathode) and a plate (anode). Following the direction of electron flow, these electrodes were identified as "A" and "B", respectively and thus the associated batteries were referred to as the "A" battery and "B" battery, respectively. Later, when the control grid element was added to create the triode tube, it was logically assigned the letter "C" and supplied from a "C" battery. Subsequent addition of further internal elements to improve the performance of the triode, did not require an extension to this series of batteries - these elements were either resistively-biased from the existing batteries, connected to ground or to the cathode.
An A battery is any battery used to provide power to the filament of a vacuum tube. It is sometimes colloquially referred to as a "wet battery" (although there's no reason why a "dry" battery of suitable voltage couldn't be utilised for the purpose; the A battery in the photo is a dry battery.) The term comes from the days of valve (tube) radios when it was common practice to use a dry battery for the plate (Anode) voltage and a rechargeable lead/acid "wet" battery for the filament voltage. (The filaments in vacuum tubes consumed much more current than the anodes, and so the "A" battery would drain much more rapidly than the "B" battery; therefore, using a rechargeable "A" battery in this role reduced the need for battery replacement. In contrast, a nonrechargeable "B" battery would need to replaced relatively infrequently.)
The prior existence of the "A" battery is apparently the reason why there is no single-A battery (cell) size, just AA and AAA. In devising the lettered sizes for single-cell "batteries," it seems that there was a conscious effort to avoid single-A, while the omission of "B" was likely due to its association with a much higher voltage.
The filament is primarily a heat source and therefore the A Battery supplies significant current and rapidly discharges. The B battery experiences comparatively little current draw and retains its stored capacity far longer than an A Battery. The voltage on a B Battery was typically 90V DC and often tapped to provide a lower voltage for screen grids or other purposes.
Even when the plate voltage rail is fed by a power supply rather than a battery, it is generally referred to as the "B+" line.
The prior existence of the "B" battery is apparently the reason why there is no B-size single-cell "battery." In devising the lettered sizes it seems that there was a conscious effort to avoid "B," possibly also due to its association with a much higher voltage. Single-A was also avoided, but there was apparently no concern for possible confusion with the use of "C" as a size.
The much higher available voltage of B batteries means that they must be handled more carefully than other battery types due to their ability to shock and/or burn the person handling them. With common flashlight cells, the normal resistance of the human body is far too high for much current to flow, but the shock hazard increases as the voltage increases.
Grid bias batteries are still manufactured today, but not for radio use. They are popular in schools and colleges as a convenient variable voltage source in science classes. The most popular battery is the 9 volt type with taps every 1½ volts that accept a banana plug.
The "C" (grid) battery is not to be confused with the C cell. In devising the lettered sizes for single-cell "batteries," it seems that there was a conscious effort to avoid "B" due to its association with a much higher voltage, and single-A was also avoided, but there was apparently no concern for possible confusion with the use of "C" as a size. This may have been because C batteries typically had voltages below 10 volts (as low as 1.5 V, the voltage of modern "C" size cells), and because C batteries typically had multiple taps, making them obviously distinct from the modern "C" size cell, which has only two terminals.