This has various names: double-double, quad-double (etc.) arithmetic; floating-point expansions. It's definitely the best way to do arithmetic at precision up to a couple of hundred digits on modern hardware, though traditional arbitrary-precision arithmetic wins at higher precision (and in any case becomes necessary, due to the exponent range).

It's unfortunate that this isn't something that was standardized and more widely available a long time ago. By contrast, IEEE binary128 and binary256 are practically useless due to lack of hardware support.