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Numerical codes that require arbitrary precision floating point (APFP) numbers for their core computation are dominated by elementary arithmetic operations due to the super-linear complexity of multiplication in the number of mantissa bits. APFP computations on conventional software-based architectures are made exceedingly expensive by the lack of native hardware support, requiring elementary operations to be emulated using instructions operating on machine-word-sized blocks. In this work, we show how APFP multiplication on compile-time fixed-precision operands can be implemented as deep FPGA pipelines with a recursively defined Karatsuba decomposition on top of native DSP multiplication. When comparing our design implemented on an Alveo U250 accelerator to a dual-socket 36-core Xeon node running the GNU Multiple Precision Floating-Point Reliable (MPFR) library, we achieve a 9.8x speedup at 4.8 GOp/s for 512-bit multiplication, and a 5.3x speedup at 1.2 GOp/s for 1024-bit multiplication, corresponding to the throughput of more than 351x and 191x CPU cores, respectively. We apply this architecture to general matrix-matrix multiplication, yielding a 10x speedup at 2.0 GOp/s over the Xeon node, equivalent to more than 375x CPU cores, effectively allowing a single FPGA to replace a small CPU cluster. Due to the significant dependence of some numerical codes on APFP, such as semidefinite program solvers, we expect these gains to translate into real-world speedups. Our configurable and flexible HLS-based code provides as high-level software interface for plug-and-play acceleration, published as an open source project.