Not really - the initial base needs to be attached to a substrate (e.g. CPG) or support. It's only removed from this substrate at the final step once you have finished extending the chain.

Think of the support as the thing that keeps your DNA in the bottle when you're doing the chemistry; during the synthesis you're repeatedly adding chemicals then washing them away.

If your DNA product isn't firmly attached to something during this process, it's going to get washed away too.

It's conceivable you could remove the DNA from the support, capture it, amplify it with PCR, then reattach to support to continue the extension, but the re-attach part would be very difficult - you'd be dealing with a long floppy chain and trying to attach one end to a solid anchor via some unknown complex chemistry. There would be side-products, loops, breaks, etc. to deal with. Someone has probably tried it, but it's not something I've ever encountered.

The initial end is bonded to a "support" (glass beads called cpg are typical) and only cleaved from the support once the entire chain is grown. The cleaving process also caps this end.

It's essentially done with robotic pipettes, though the ones used in DNA synthesis are very specialized.

It can be done with hand pipettes, but it would be a slow and error prone process.

The materials are not exotic and can be purchased from chemical supply.

Typically the starting material is a (population of) a single DNA (or RNA) base bonded to a glass bead (cpg), and the chemistry is done by washing the beads with a series of chemicals, following a repeated recipe. The recipe is identical for each base added to the chain (excepting the base itself of course).

The practical limit is a multiplicative function of the error rate of the chemistry involved.

If the %succes of adding a new base to the growing chain is X, and the length of the chain is N, the overall % of success for a given length is X raised to the power of N.

So for example;

If you have a 98% success rate of adding a new base to the growing chain (ie 98% of the millions of chains you are extending successfully extend),

by the time you are adding base 100, you have 98%^100 or ~13% of the original starting material as 100 base long oligonucleotides.

The shorter chains are discarded during a purification step after you've finish the addition of bases.

Practical limit of synthesis length is usually under 100 for this reason, but it is then possible (as someone mentioned in another comment) to join these together in a separate chemical process (ligation, not to be confused with the surgical definition)