It's common to think about repetitive waveforms but that doesn't have to be the case for an 'arbitrary' waveform. The waveform will repeat but the experiment may be over before it does.
In the case of the Siglent SDG2082X, the device supports sample memory from 8 points up to 8 megapoints and sample rates from 1 uSa/s up to 75MSa/s in what they call TrueArb mode. In DDS mode the sample rate is 300 MSa/s. And then they claim a sample rate of 1.2GSa/s using 4x interpolation.
TrueArb mode is point by point, every output voltage comes from sample memory.
DDS will generate most common waveforms and most AWGs have a large menu of built-in waveforms. TrueArb is different in that there are discrete samples for every time tick.
There is some very useful iinformation in the SDG2000 datasheet:
https://mediacdn.eu/mage/media/downloads/SDG2000X_DataSheet.pdfAbout page 6, they talk about the EasyWave software which allows for manual drawing, line-drawing, equation-drawing, coordinate-drawing and so forth.
User Manual:
https://mediacdn.eu/mage/media/downloads/SDG2000X_UserManual.pdfStarting on page 40, there is a list of built-in functions extending for 4 pages. There are a LOT of built-ins.