replay does not consider a Constant referenced by an Expression referenced by the Functional

Yes, the Expressions have the wrong .t's in the UFL forms for the time integration inside the functional.

A hack workaround, for now:

from dolfin import *
from dolfin_adjoint import *

t = Constant(0.0, name="t")

mesh = UnitCubeMesh(1,1,1)
d = 3
V = VectorFunctionSpace(mesh, "CG", 1)
R = FunctionSpace(mesh, "R", 0)

ut = TrialFunction(V)
v = TestFunction(V)

u = Function(V, name="u")
g = Function(V, name="g")

c = Constant(0.0, name="c")
f = as_vector((c, 0.0, 0.0))
a = dot(ut,v)*dx
L = dot(f,v)*dx
solve(inner(ut, v)*dx == inner(u, v)*dx, u) # initial condition annotation

z = Function(R)
rtest = TestFunction(R)
rtrial = TrialFunction(R)
z_expr = Expression("1.0 + 1.0*t", t=t)
za = inner(rtrial, rtest)*dx
zL = inner(z_expr, rtest)*dx
solve(za == zL, z)

# A timeloop that assigns to t but doesn't use it in the computation of u
t.assign(0.0)
u.interpolate(Expression(("0.0",)*d))
adj_start_timestep(0.0)
for i in range(1,3):
    t.assign(i)
    solve(a == L, u)
    solve(za == zL, z)
    print "time == %s, z == %s" % (i, z.vector()[0])
    adj_inc_timestep(i, finished=i==2)

# Validates the replay of u only
assert replay_dolfin(forget=False)

def on_replay(var, func, m):
    print "Replay at %s of %s, ||%s|| = %g" % (var.timestep, str(var), func.name(), norm(func))

adj_html("forward.html", "forward")

# A functional that references t through an Expression attribute
Jdist = (z**2)*dx*dt
Jfunc = Functional(Jdist)
Jred = ReducedFunctional(Jfunc, [ScalarParameter(c)], replay_cb=on_replay)
print Jred([c])

prints 9.0 as expected.