The serial number is in the battery compartment.
Some older DD-3s have a larger, postage stamp-sized chip, similar to the older DD-2. These are supposed to sound "better".
From the Boss Area site:
"The early DD-3 is the same pedal as the DD-2. Because of the price drop on DRAM in 1986 it was possible to produce the pedal much cheaper. Dropping the price didn't look so good so it was decided to re-release the DD-2 under the name DD-3 instead. The front of the pedal got a face lift with the re-release.
Version differences
There are 3 main versions of the DD-3. The first version was released 1986 but it was redisigned with a new DSP gate array chip December 1990. The new version was called DD-3A. The DC socket was at the same time replaced with the smaller PCB mounted type and this can be used to identify the version. The design change happened before production was moved from Japan to Taiwan so all DD-3s made in Taiwan are DD-3A or DD-3B versions. The latest version of the DD-3 was introduced August 2001 and is called DD-3B. The DD-3B uses a new type of DRAM as well as surface mounted components. The PCB layout is therefore radically changed compared to the DD-3A."
DD-3
Approx. Serial: 690000 - NC09999
Marking of PCB: 2291088702
Adaptor jack: Large Hosiden
Main Controller: Roland RDD63H101
DRAM: HM4864P-3, M5K4164ANP or MSM3764-20RS
DD-3A
Serial ZC10000 - SO91199
Marking of PCB: 2292098400
Adaptor jack: small PCB Mounted
Main Controller: MN51010RBA
DRAM: M5M4464-L12
DD-3B
Serial SO91200 - present
Marking of PCB: 75D563P000
Adaptor jack: small PCB Mounted
Main Controller: MN51010RBA BOS-008
DRAM: M11B416256A-35J(T)
Whatever. YMMV.