Inclusion Criteria:

- Diagnosis meets one of the following criteria:

- Part A (both schedules):

- Patients must have a diagnosis of recurrent or refractory solid tumors,
including CNS tumors or lymphoma

- Patients must have had histologic verification of malignancy at original
diagnosis or relapse except in patients with intrinsic brain stem tumors,
optic pathway gliomas, or patients with pineal tumors and elevations of
cerebrospinal fluid (CSF) or serum tumor markers, including
alpha-fetoprotein or beta-human chorionic gonadotropin (hCG)

- Part B (both schedules):

- Patients must have a diagnosis of recurrent or refractory leukemia

- Patients with solid tumors must have either measurable or evaluable disease

- Patients with leukemia must have >= 5% blasts in the bone marrow

- Active extramedullary disease (except for leptomeningeal disease) may also be
present

- Patient's current disease state must be one for which there is no known curative
therapy or therapy proven to prolong survival with an acceptable quality of life

- Slides or tissue blocks from either initial diagnosis or relapse must be available
for central review (if tissue blocks or slides are unavailable, the study chair must
be notified prior to study enrollment)

- Karnofsky >= 50% for patients > 16 years of age and Lansky >= 50% for patients =< 16
years of age

- Neurologic deficits in patients with CNS tumors must have been relatively stable
for a minimum of 1 week prior to study enrollment; patients who are unable to
walk because of paralysis, but who are up in a wheelchair, will be considered
ambulatory for the purpose of assessing the performance score

- For patients with solid tumors without known bone marrow involvement including
patients who are status post-stem cell transplantation:

- Peripheral absolute neutrophil count (ANC) >= 1,000/mm^3

- Platelet count >= 100,000/mm^3 (transfusion independent, defined as not
receiving platelet transfusions within a 7-day period prior to enrollment)

- For patients with solid tumors with known bone marrow metastatic disease:

- These patients are eligible for study provided they meet the blood count
criteria above and are not known to be refractory to red cell or platelet
transfusions

- For patients with leukemia (part B):

- Blood counts are not required to be normal prior to enrollment on this trial,
however, platelet count has to be >= 20,000/mm^3 (may receive platelet
transfusions)

- Creatinine clearance or radioisotope glomerular filtration rate (GFR) >= 70 mL/min^2
OR a serum creatinine based on age/gender as follows:

- =< 0.6 mg/dL (for patients 1 to < 2 years old)

- =< 0.8 mg/dL (for patients 2 to < 6 years old)

- =< 1 mg/dL (for patients 6 to < 10 years old)

- =< 1.2 mg/dL (for patients 10 to < 13 years old)

- =< 1.4 mg/dL (for female patients >= 13 years old)

- =< 1.5 mg/dL (for male patients 13 to < 16 years old)

- =<1.7 mg/dL (for male patients >= 16 years old)

- Bilirubin (sum of unconjugated plus conjugated) =< 1.5 x upper limit of normal (ULN)
for age

- Serum glutamate pyruvate transaminase (SGPT) (aspartate aminotransferase [ALT]) =<
110 U/L for patients with solid tumors OR SGPT (ALT) =< 225 U/L for patients with
leukemias (for the purpose of this study, the ULN for SGPT is 45 U/L)

- Serum albumin >= 2 g/dL

- QTc =< 450 msec

- Not pregnant or nursing

- Negative pregnancy test

- Fertile patients must use effective contraception

- Patients must be able to swallow whole tablets

- Nasogastric or gastrostomy (G)-tube administration is not allowed

- Patients must have a body surface area (BSA) > 0.5 m^2 when enrolling on dose levels
0 or 1 of the every other day schedule

- No BSA restrictions apply to patients enrolling on higher dose levels

- No BSA restrictions apply to patients enrolling on any dose level of the weekly
schedule

- Patients with seizure disorder may be enrolled if on non-enzyme-inducing
anticonvulsants and well controlled

- Nervous system disorders (Common Terminology Criteria for Adverse Events [CTCAE] v4)
resulting from prior therapy must be =< grade 2

- Patients who have an uncontrolled infection are not eligible

- Patients with known type I or type II diabetes mellitus are not eligible

- Patients who, in the opinion of the investigator, may not be able to comply with the
safety monitoring requirements of the study are not eligible

- Patients must have fully recovered from the acute toxic effects of all prior
anti-cancer chemotherapy

- Patients with solid tumors must not have received myelosuppressive chemotherapy
within 3 weeks of enrollment onto this study (6 weeks if prior nitrosourea)

- Patients with leukemia who relapse while receiving standard maintenance chemotherapy
will not be required to have a waiting period before enrollment onto this study

- Patients with leukemia who relapse while they are not receiving standard maintenance
therapy, must have fully recovered from all acute toxic effects of prior therapy

- At least 14 days must have elapsed since the completion of cytotoxic therapy,
with the exception of hydroxyurea

- Cytoreduction with hydroxyurea can be initiated and continued for up to 24 hours
prior to the start of MK-2206

- At least 14 days after the last dose of a long-acting growth factor (e.g., Neulasta)
or 7 days for short-acting

- For agents that have known adverse events occurring beyond 7 days after
administration, this period must be extended beyond the time during which
adverse events are known to occur (the duration of this interval must be
discussed with the study chair)

- At least 7 days after the last dose of a biologic agent

- For agents that have known adverse events occurring beyond 7 days after
administration, this period must be extended beyond the time during which
adverse events are known to occur (the duration of this interval must be
discussed with the study chair)

- At least 6 weeks since the completion of any type of immunotherapy (e.g., tumor
vaccines)

- At least 3 half-lives of the antibody after the last dose of a monoclonal antibody

- At least 2 weeks for local palliative radiation therapy (XRT) (small port); >= 24
weeks must have elapsed if prior total body irradiation (TBI), craniospinal XRT, or
if >= 50% radiation of pelvis; >= 6 weeks must have elapsed if other substantial bone
marrow (BM) radiation

- No evidence of active graft vs host disease and >= 8 weeks must have elapsed since
transplant or stem cell infusion without TBI

- At least 3 months since bone marrow transplantation

- Patients receiving corticosteroids who have not been on a stable or decreasing dose
of corticosteroid for the prior 7 days are not eligible

- Patients who are currently receiving another investigational drug are not eligible

- Patients who are currently receiving other anticancer agents are not eligible (except
leukemia patients receiving hydroxyurea, which may be continued until 24 hours prior
to start of protocol therapy)

- Patients with leukemia may receive intrathecal therapy

- Patients must not be receiving enzyme-inducing anticonvulsants

- Patients receiving insulin or growth hormone therapy are not eligible

- Patients on medications that may cause QTc interval prolongation are not eligible

- Patients who are receiving cyclosporine, tacrolimus, or other agents to prevent
either graft-versus-host disease post-bone marrow transplant or organ rejection
post-transplant are not eligible for this trial

- Other concurrent cancer therapy, including chemotherapy, radiation therapy,
immunotherapy, or biologic therapy may NOT be administered to patients receiving
study drug