You can use something other than the standard R-2R ladder, but there are drawbacks. The reason R-2R is such a win is that instead of having to assemble a wide range of accurate resistor values, each trimmed to meet some logarithmic current per step spec, the R-2R ladder only needs to have two values: R and 2R, and the 2R can be made of one R and another R in series. In short, all you need is to be able to make one resistor repeatably, with a value somewhat close to what you really want, maybe ±20%.

This is why PMI's DAC-08 chip of the late 70s was able to be fabricated cheaply on an ancient monolithic process and still have excellent monotonicity, despite the difficulty of implanting a precise resistor onto an IC. The ladder accuracy was not dependent on the deposition of the resistor, which determines its absolute value, but instead upon the accuracy of the lithography, which determines the resistor ratio accuracy. So, while the DAC-08's ladder might have resistors whose absolute value ranged around ±20%, their ratio accuracy was around 1 part in 1000, enough for 1/4 bit of an 8 bit DAC.

I suppose that modern lithography could make some fairly precise "special" ratios on a modern chip, so that one could implement a log current per switch code ladder, but given a resistor implantation, the values can't have too wide of a range without making some resistors really large and some really small. This would make the small resistors less accurate, and the big ones too expensive in terms of die area.

Finally, dividing up a large problem into a logarithmic problem, which is basically what an R-2R ladder does, tends to make the circuit smaller and more minimal.

So... one can still have a logarithmic switch code with an R-2R ladder, as long as you add a layer of logic to translate the logarithmic input code to the native linear code used by the actual ladder. IIRC, Analog Devices made some LogDACs in the 80s that were just that - a competent R-2R ladder (or a few of them), connected with logic to achieve a dB per step code.

To this day, monolithic R-2R ladders seem to be limited to around 12, maybe 13 bits per ladder, and if you want more bits than that, several of these ladders can be lashed together, perhaps with calibration or trimming to get the commercially available "wide" multiplying DACs. You can buy precision resistor arrays however and do your own DAC switching, and in that world, the sky's the limit for accuracy and price.

I'm not trying to poo-poo the idea of a log ladder, but in practice, it's a real pain to get that right, especially compared to a straight, linear, R-2R ladder. The expense is basically the resistors, and having to trim a huge pile of values to get a log ladder is expensive and hard to stabilize. If you spent the same money on a pile of "one value" resistors, they might actually track over temperature and time, and would be cheaper overall, despite the fact that you'd need more of them.